Yes, you do. If you don't need a serial number to install XP, then you must have 3 computers all from the same manufacturer. To upgrade them all to Windows 7, you must purchase 3 licenses... one for each. The reason you're able to get away with one copy of XP is because each computer already has a license for it and since they're likely from the same manufacturer, the same OEM CD works for all of them without asking for a product key.
Regardless of whether you go Ultimate, or with the Home Premium family pack as others have suggested, you will need 3 licenses (the family pack is essentially 3 serials in one package). If there is a feature in Ultimate that you must have, then go for it. Otherwise, save yourself some cash and get a lower SKU.