I have got the new iOS 4.1 Game Center working almost perfectly on iPhone 3G, running iOS 4.1. All thanks to the latest Redsn0w 0.9.6 which makes jailbreaking possible on iPhone 3G, running iOS 4.1.
As you can see it in the video embedded below, iPhone 3G is quite capable of running Game Center, yet Apple decided against it for iPhone 3G users running iOS 4.1.

The process involves modification of some .plist files found in the original GameCenter.ipa file for iPhone 3GS, and then SSH’ing this modified file on to jailbroken iPhone 3G running iOS 4.1. Since modifying .ipa files breaks Apple’s DRM, I cant share the exact instructions, or the modified file with you.

Without any further delay, here is the video demonstration of the new iOS 4.1 Game Center running on iPhone 3G, alongside iPhone 4 and iPhone 3GS.