Why am I unable to use Game Center on a jail broken phone?
iPhone 4 GSM Black:

I had the same issue when I updated to 6.1.2 from my previous version 5.1.1 when the evasi0n jailbreak came out. I solved the problem by restoring to the factory default. Then I went through the beginning steps of the iphone introduction, when i open and logged in to gamecenter everything came through pretty smooth... I jailbreaked it after I restored  and gamecenter still worked smoothly. Then I restored to my last back-up and opened gamecenter after it all loaded up and everything is still working fine.
Hopefully this simple troubleshooting helped out. It did for me.