My games tend to have a medieval or bronze age feel; I basically require everyone to have a god, cleric or not. I understand that some people like the idea of playing modern people in pre-modern worlds, in which case godless clerics would be quite reasonable (we elected one to our city council here in Vancouver a year ago) but it's just not my thing.
And no. I don't let monks into my games either.
And no. I don't let monks into my games either.