It took me some time to find a solution to this problem, but here is a working solution.

 I wanted to be able to drag 'n drop WPF DataGrid rows in MVVM, and although this solution is not strickly MVVM, I still think that it fits nicely in that design patten. All GUI it handled within the view, and all logic in the ViewModel.

 I found a good starting point for this solution somewhere, but again unfortunately I have forgotten where. I think it was on codeproject. My sincere apologies to the author.

 The follwing goes into the UserControl code behind:

using Baltz.ViewModel;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.Windows.Documents;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Media.Imaging;
using System.Windows.Navigation;
using System.Windows.Shapes;

namespace Baltz.View
{
    /// <summary>
    /// Interaction logic for DataGridDraggableRowView.xaml
    /// </summary>
    public partial class DataGridDraggableRowView : UserControl
    {
        public DataGridDraggableRowView()
        {
            InitializeComponent();
        }

        #region Static Methods
        private static T FindVisualParent<T>(UIElement element) where T : UIElement
        {
            var parent = element;
            while (parent != null)
            {
                var correctlyTyped = parent as T;
                if (correctlyTyped != null)
                {
                    return correctlyTyped;
                }

                parent = VisualTreeHelper.GetParent(parent) as UIElement;
            }
            return null;
        }
        #endregion

        #region Fields
        private int _originalIndex;
        private DataGridRow _oldRow;
        private DataGridDraggableRowItem _targetItem;
        private DataGridDraggableRowViewModel _viewModel;
        #endregion Fields

        #region Event Handlers
        /// <summary>
        /// Updates the grid as a drag progresses
        /// </summary>
        private void OnMainGridCheckDropTarget(object sender, DragEventArgs e)
        {
            var row = FindVisualParent<DataGridRow>(e.OriginalSource as UIElement);

            // Set the DragDropEffects 
            if ((row == null) || !(row.Item is DataGridDraggableRowItem))
            {
                e.Effects = DragDropEffects.None;
            }
            else
            {
                var currentIndex = row.GetIndex();

                // Erase old drop-line
                if (_oldRow != null)
                    _oldRow.BorderThickness = new Thickness(0);

                // Draw new drop-line
                var direction = (currentIndex - _originalIndex);
                if (direction < 0)
                    row.BorderThickness = new Thickness(0, 2, 0, 0);
                else if (direction > 0)
                    row.BorderThickness = new Thickness(0, 0, 0, 2);
                row.BorderBrush = new SolidColorBrush(Color.FromArgb(0xFF, 0x00, 0x9C, 0xDD));
                // Reset old row
                _oldRow = row;
            }
        }

        /// <summary>
        /// Gets the view model from the data Context and assigns it to a member variable.
        /// </summary>
        private void OnMainGridDataContextChanged(object sender, DependencyPropertyChangedEventArgs e)
        {
            _viewModel = (DataGridDraggableRowViewModel)this.DataContext;
        }

        /// <summary>
        /// Process a row drop on the DataGrid.
        /// </summary>
        private void OnMainGridDrop(object sender, DragEventArgs e)
        {
            e.Effects = DragDropEffects.None;
            e.Handled = true;

            // Verify that this is a valid drop and then store the drop target
            var row = FindVisualParent<DataGridRow>(e.OriginalSource as UIElement);
            if (row != null)
            {
                _targetItem = row.Item as DataGridDraggableRowItem;
                if (_targetItem != null)
                {
                    e.Effects = DragDropEffects.Move;
                }
            }

            // Erase last drop-line
            if (_oldRow != null)
                _oldRow.BorderThickness = new Thickness(0, 0, 0, 0);
        }

        /// <summary>
        /// Processes a drag in the main grid.
        /// </summary>
        private void OnMainGridMouseMove(object sender, MouseEventArgs e)
        {
            // Exit if left mouse button aren't pressed
            if (e.LeftButton != MouseButtonState.Pressed)
                return;

            // Find the row the mouse button was pressed on
            var row = FindVisualParent<DataGridRow>(e.OriginalSource as FrameworkElement);
            _originalIndex = row.GetIndex();

            // If the row was already selected, begin drag
            if ((row != null) && row.IsSelected)
            {
                // Get the grocery item represented by the selected row
                var selectedItem = (DataGridDraggableRowItem)row.Item;
                var finalDropEffect = DragDrop.DoDragDrop(row, selectedItem, DragDropEffects.Move);
                if ((finalDropEffect == DragDropEffects.Move) && (_targetItem != null))
                {
                    /* A drop was accepted. Determine the index of the item being 
                     * dragged and the drop location. If they are different, then 
                     * move the selectedItem to the new location. */

                    // Move the dragged item to its drop position
                    var oldIndex = _viewModel.Rows.IndexOf(selectedItem);
                    var newIndex = _viewModel.Rows.IndexOf(_targetItem);
                    if (oldIndex != newIndex)
                        _viewModel.Rows.Move(oldIndex, newIndex);
                    _targetItem = null;
                }
            }
        }
        #endregion
    }
}

A side note: This style is based on a style posted in another post

<ResourceDictionary x:Class="Ellab.Main.Settings.SettingsWindow.View.Styles.DraggableDataGridStyle"
                    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
                    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
                    xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006" 
                    xmlns:d="http://schemas.microsoft.com/expression/blend/2008" 
                    mc:Ignorable="d">
    <ResourceDictionary.MergedDictionaries>
        <ResourceDictionary Source="DataGridStyle.xaml"/>
    </ResourceDictionary.MergedDictionaries>
    <Style x:Key="DefaultDraggableDataGrid" BasedOn="{StaticResource DefaultSettingsDataGrid}" TargetType="{x:Type DataGrid}">
        <Setter Property="HeadersVisibility" Value="Column"/>
        <Setter Property="AllowDrop" Value="True"/>
    </Style>
    <Style TargetType="{x:Type DataGridCell}">
        <Style.Resources>
            <SolidColorBrush x:Key="{x:Static SystemColors.HighlightBrushKey}" Color="#FF009CDD" />
            <SolidColorBrush x:Key="{x:Static SystemColors.ControlBrushKey}" Color="#FF009CDD" />
            <SolidColorBrush x:Key="{x:Static SystemColors.HighlightTextBrushKey}" Color="Black" />
            <SolidColorBrush x:Key="{x:Static SystemColors.ControlTextBrushKey}" Color="Black" />
        </Style.Resources>
    </Style>
</ResourceDictionary>

And this is the complete View - most of it isn't relevant for the dragable rows.

<UserControl x:Class="foo.View.barDataGridDraggableRowView"
             xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
             xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
             xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006" 
             xmlns:d="http://schemas.microsoft.com/expression/blend/2008" 
             mc:Ignorable="d" 
             d:DesignHeight="300" d:DesignWidth="300">
    <UserControl.Resources>
        <ResourceDictionary>
            <ResourceDictionary.MergedDictionaries>
                <ResourceDictionary Source="Styles/DraggableDataGridStyle.xaml"/>
                <ResourceDictionary Source="Styles/ContentPresenterStyle.xaml"/>
            </ResourceDictionary.MergedDictionaries>
        </ResourceDictionary>
    </UserControl.Resources>
    <StackPanel Orientation="Vertical" Background="Transparent">
        <ContentPresenter Content="{Binding Path=HeaderButton}" Style="{StaticResource DefaultContentPresenter}"/>
        <StackPanel Orientation="Vertical" Background="Transparent" Visibility="{Binding Path=IsExpanded}">
            <DataGrid ItemsSource="{Binding Path=Rows}"
                  MouseMove="OnMainGridMouseMove"
                  DragEnter="OnMainGridCheckDropTarget"
                  DragLeave="OnMainGridCheckDropTarget"
                  DragOver="OnMainGridCheckDropTarget"
                  Drop="OnMainGridDrop" 
                  DataContextChanged="OnMainGridDataContextChanged"
                  Style="{StaticResource DefaultDraggableDataGrid}">
                <DataGrid.Columns>
                    <DataGridTemplateColumn IsReadOnly="True" Width="20">
                        <DataGridTemplateColumn.CellTemplate>
                            <DataTemplate>
                                <Image Source="../../Resources/DragDrop.png" Stretch="Fill" Opacity="{Binding Opacity}"/>
                            </DataTemplate>
                        </DataGridTemplateColumn.CellTemplate>
                    </DataGridTemplateColumn>
                    <DataGridTemplateColumn IsReadOnly="True">
                        <DataGridTemplateColumn.Header>
                            <TextBlock Text="{Binding DataContext.EnabledColumnHeader, RelativeSource={RelativeSource AncestorType=DataGrid}}" MaxWidth="65" TextWrapping="WrapWithOverflow" TextTrimming="CharacterEllipsis">
                                <TextBlock.LayoutTransform>
                                    <TransformGroup>
                                        <RotateTransform Angle="90" />
                                        <ScaleTransform ScaleX="-1" ScaleY="-1"/>
                                    </TransformGroup>
                                </TextBlock.LayoutTransform>
                            </TextBlock>
                        </DataGridTemplateColumn.Header>
                        <DataGridTemplateColumn.CellTemplate>
                            <DataTemplate>
                                <CheckBox HorizontalAlignment="Center" VerticalAlignment="Center" IsChecked="{Binding EnabledChecked, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay}" Opacity="{Binding Opacity}" IsEnabled="{Binding Enable}"/>
                            </DataTemplate>
                        </DataGridTemplateColumn.CellTemplate>
                    </DataGridTemplateColumn>
                    <DataGridTemplateColumn IsReadOnly="True">
                        <DataGridTemplateColumn.Header>
                            <TextBlock Text="{Binding DataContext.WizardColumnHeader, RelativeSource={RelativeSource AncestorType=DataGrid}}" MaxWidth="65" TextWrapping="WrapWithOverflow" TextTrimming="CharacterEllipsis">
                                <TextBlock.LayoutTransform>
                                    <TransformGroup>
                                        <RotateTransform Angle="90" />
                                        <ScaleTransform ScaleX="-1" ScaleY="-1"/>
                                    </TransformGroup>
                                </TextBlock.LayoutTransform>
                            </TextBlock>
                        </DataGridTemplateColumn.Header>
                        <DataGridTemplateColumn.CellTemplate>
                            <DataTemplate>
                                <CheckBox HorizontalAlignment="Center" VerticalAlignment="Center" IsChecked="{Binding WizardChecked, UpdateSourceTrigger=PropertyChanged, Mode=TwoWay}" Opacity="{Binding Opacity}" IsEnabled="{Binding Enable}" Visibility="{Binding WizardVisible}"/>
                            </DataTemplate>
                        </DataGridTemplateColumn.CellTemplate>
                    </DataGridTemplateColumn>
                    <DataGridTemplateColumn IsReadOnly="True" Width="*">
                        <DataGridTemplateColumn.Header>
                            <TextBlock Text="{Binding DataContext.TextColumnHeader, RelativeSource={RelativeSource AncestorType=DataGrid}}" TextWrapping="WrapWithOverflow" TextTrimming="CharacterEllipsis" VerticalAlignment="Bottom"/>
                        </DataGridTemplateColumn.Header>
                        <DataGridTemplateColumn.CellTemplate>
                            <DataTemplate>
                                <TextBlock Text="{Binding Text}" HorizontalAlignment="Left" VerticalAlignment="Center" Opacity="{Binding Opacity}" IsEnabled="{Binding Enable}"/>
                            </DataTemplate>
                        </DataGridTemplateColumn.CellTemplate>
                    </DataGridTemplateColumn>
                    <DataGridTemplateColumn IsReadOnly="True" Width="40">
                        <DataGridTemplateColumn.CellTemplate>
                            <DataTemplate>
                                <Image Source="{Binding TileImage}" Opacity="{Binding Opacity}"/>
                            </DataTemplate>
                        </DataGridTemplateColumn.CellTemplate>
                    </DataGridTemplateColumn>
                </DataGrid.Columns>
            </DataGrid>
            <ContentPresenter Content="{Binding Path=TilesRemainingText}"/>
        </StackPanel>
    </StackPanel>
</UserControl>

As always, feel free to comment, or ask.