So this has happened on numerous occasions, and I'd like to know some opinions on how people will play this hand.

You are in LP and holding KK and rais preflop, getting 2 callers.
Flop comes A-rag-rag and it is checked around and you c-bet.

Now obviously if there's a re-raise here, you're going to fold most times.
But what if it's just a call?

I have had this happen numerous times and I have no idea what to do when I don't have a read on my opponent. Would you c-bet again or check on the turn? What about on the river?

Any opinions would be appreciated.