The long-awaited patch for Fez has finally arrived, though you may want to hold the fanfare. While the patch does fix a number of glitches and prevents a lot of exploits and problematic "infinite death loops" (eek!), it also introduces a "fairly widespread" new problem, according to Polytron. Specifically, "the save file can be seen as corrupted by the game after installing the patch."
Clearing the Xbox 360 cache "should resolve the issue." If that doesn't work, however, "there is no known workaround." In other words, you probably shouldn't play Fez until Polytron patches the patch.