It's gonna be mess, no matter from what side it's approached. I'd say your best bet (provided that your grid cells content won't be wrapped) is to use a visual brush (with lines) as your DataGrid's background.
UPDATE 1 - XAML
It's alwast there, the trick is to use MinHeight, which will produce a vision of blank items thanks to tiled background. If your grid will be populated with the real data the background will expand, rendering more lines.
One thing I didn't try is how it'll handling scrolling.
Here's an example:
<Grid>
<Grid.Resources>
<VisualBrush x:Key="StripesBrush" TileMode="Tile" Viewport="0,0,5,20"
Viewbox="0,0,10,10" ViewportUnits="Absolute"
ViewboxUnits="Absolute">
<VisualBrush.Visual>
<Line X1="0" X2="10000" Y1="0" Y2="0" Stroke="DarkGray"/>
</VisualBrush.Visual>
</VisualBrush>
</Grid.Resources>
<DataGrid x:Name="g" AutoGenerateColumns="False"
GridLinesVisibility="None"
MinHeight="100"
Height="100"
VerticalAlignment="Top"
Background="{StaticResource StripesBrush}">
<DataGrid.Columns>
<DataGridTextColumn Header="A" Width="Auto" Binding="{Binding Item1}">
<DataGridTextColumn.CellStyle>
<Style TargetType="DataGridCell">
<Setter Property="Background" Value="Transparent"></Setter>
</Style>
</DataGridTextColumn.CellStyle>
</DataGridTextColumn>
<DataGridTextColumn Header="B" Width="Auto" Binding="{Binding Item2}" />
</DataGrid.Columns>
</DataGrid>
</Grid>
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