I have been having the same problem for a while now. at first it was just one game that would not update achievements. then I noticed other games not updating, or adding a few achievment's but not all that was earned. I have tried a few things and some have worked and some did not. for anyone still looking at this, I figured I will share my findings.
Restarting my idevice (4th gen 6.1.5). Nothing happened
sing in and sing out. nothing happened
Swipe the game having the problem, removing it from the game center list, while clicking cancel to not uninstall the game.(making sure the game isn't open in the background). this might work, and for me it is not a permanent fix. it would work for one launch of the game, and/or just update some achievment's.
what I just tried was a combination of the two. I made sure that none of the games(any) were opened, and closed any that was. opened game center. removed one that I knew that had a problem updating. after I restarted my ipod (regular close from lock button) after booting idevice I open game center I singed out ( could try at first attempt to make sure the game was removed) and rebooted. then I re-signed in. the game was there with 0 achievements but still had my old ones. opened the game and started playing. mid game I was asked to sing into my gc and I was able to unlock achievment's in other games as well. I did this for pocket mine. let me know if this works for. and I don't know how reliable this is.