range.Borders(Excel.XlBordersIndex.xlEdgeLeft).LineStyle = Excel.XlLineStyle.xlLineStyleNone
range.Borders(Excel.XlBordersIndex.xlEdgeRight).LineStyle = Excel.XlLineStyle.xlLineStyleNone
range.Borders(Excel.XlBordersIndex.xlEdgeTop).LineStyle = Excel.XlLineStyle.xlLineStyleNone
range.Borders(Excel.XlBordersIndex.xlEdgeBottom).LineStyle = Excel.XlLineStyle.xlLineStyleNone
range.Borders(Excel.XlBordersIndex.xlInsideHorizontal).LineStyle = Excel.XlLineStyle.xlLineStyleNone
range.Borders(Excel.XlBordersIndex.xlInsideVertical).LineStyle = Excel.XlLineStyle.xlLineStyleNone
Removes the borders around the cells and between cells (via xlInsideHorizontal and xlInsideVertical). If you expect diagonal borders, include xlDiagonalDown and xlDiagonalUp.
Okay, the above code was very verbose. The following should do it too:
For Each border in range.Borders
border.LineStyle = Excel.XlLineStyle.xlLineStyleNone
Next
See: http://msdn.microsoft.com/en-us/library/microsoft.office.interop.excel.borders.aspx