Well, if you actually do that, you won't be able to sort on the columns with the duplicates any more.
Conditional formatting technique
I'd suggest you use conditional formatting to hide them instead of deleting them. You'll want to select the whole region that you want to hide duplicates in. Look at which cell is active within the selection, it should say the address to the left of the formula bar.
Go to the format menu, choose Conditional Formatting. In the dialog, choose "formula." Now, write a formula like =A2=A1, only A2 should be the cell you selected and A1 should be the cell above it. Then use the formatting options in the dialog to make the text invisible.
Formula / paste special technique
If you really want to delete the blanks, you can try this approach. Insert columns next to the columns you want to delete duplicates from. Type in a formula =IF(A2=A1, "", A2), using the same rule as above. Copy that formula into all appropriate cells. Then select those cells and use Paste Special... to paste just the values in. As I mentioned earlier, sorting won't work any more with this technique.