If your games saves were made using a 'Live' account, then you will need to use the same live account to access the game saves. You can have an offline live account, you just need to re-download you gamertag to your memory chip. (you may have to temporarily connect your Xbox to internet to do so). In the future you can just keep your gamertag on your memory card, and that way it will be portable along with your game saves. You can still get achievements, but there will be no date stamp, and it will not register with xbox.com until you are connected to internet.
I do not think an offline account with the same gamertag name will be able to access the game saves made by the live account.
Xbox and PS3 Gamertag = Xerostomia
Wii = I'm too lazy to get the number