Have a look at the following:

http://www.cpearson.com/excel/RowLiner.htm

Not exactly what you want but very useful.

Hi Michael,

Unfortunately changing cells in the event cancels cut/copy. Perhaps disable
changes if in cut or copy mode -

Private Sub Worksheet_SelectionChange(ByVal Target As Range)
Static OldRange As Range

If Application.CutCopyMode = 0 Then

   If Not OldRange Is Nothing Then
       OldRange.EntireRow.Interior.ColorIndex = xlNone
   End If

   Set OldRange = Target(1).EntireRow
   OldRange.Interior.ColorIndex = 36
Else
   If OldRange Is Nothing Then
       Set OldRange = Target(1).EntireRow
   Else
   Set OldRange = Union(OldRange, Target(1).EntireRow)
   End If
End If

End Sub

Apart from disabling I've made some other changes to your routine which you
may or may not prefer - previously selecting another cell in same row left
the row un-coloured (another way is first remove format from all rows in the
used range and not bother with "OldRange").

This is far from satisfactory but might be enough for your needs. I wouldn't
use this myself! (though I do use something very vaguely along the lines of
Chip Pearson's RowLiner as referred to you by "bigwheel").

Regards,
Peter T