@nutsch’s answer is perfectly fine and should work 99.9% of the time. In the rare occasion that FSO is not available, here’s a version that doesn’t have a dependency. As is, it does require that the source worksheet doesn’t have any blank rows in the content section.
Sub SaveRowsAsCSV()
Dim wb As Excel.Workbook, wbNew As Excel.Workbook
Dim wsSource As Excel.Worksheet, wsTemp As Excel.Worksheet
Dim r As Long, c As Long
Set wsSource = ThisWorkbook.Worksheets("worksheet")
Application.DisplayAlerts = False 'will overwrite existing files without asking
r = 1
Do Until Len(Trim(wsSource.Cells(r, 1).Value)) = 0
ThisWorkbook.Worksheets.Add ThisWorkbook.Worksheets(1)
Set wsTemp = ThisWorkbook.Worksheets(1)
For c = 2 To 7
wsTemp.Cells((c - 1) * 2 - 1, 1).Value = wsSource.Cells(r, c).Value
Next c
wsTemp.Move
Set wbNew = ActiveWorkbook
Set wsTemp = wbNew.Worksheets(1)
'wbNew.SaveAs wsSource.Cells(r, 1).Value & ".csv", xlCSV 'old way
wbNew.SaveAs "textfile" & r & ".csv", xlCSV 'new way
'you can try other file formats listed at http://msdn.microsoft.com/en-us/library/office/aa194915(v=office.10).aspx
wbNew.Close
ThisWorkbook.Activate
r = r + 1
Loop
Application.DisplayAlerts = True
End Sub