It would sound better if you said "I went to the dentist to have my teeth cleaned" 'have' could be replaced by 'get'

I think the most colloquial American way would be: I went to the dentist for a cleaning today. We wouldn't say hospital and teeth would be made obvious by the fact that we went to the dentist.

"TO CLEAN TEETH" sounds like YOU are the one who is cleaning teeth. :) "FOR TEETH CLEANING" is correct, but I would add "a teeth cleaning." I would say, "I went to the hospital (or dentist's) for a teeth cleaning yesterday."