Conceptual Problem with AK/Turn check-raise.

[Ortho]

This hand raised something that has really been bothering me.

I've only been at this table for about 4 hands, and I have no read on villain. My flop raise could well be incorrect, but I think it's reasonable to push UTG+1 out and possibly take a free card on the turn, so please ignore it if you're willing to because my question concerns the turn.

UTG folds, UTG+1 limps, Villain raises, 3 players fold, Hero 3-bets in CO with A[img]/images/graemlins/spade.gif[/img]K[img]/images/graemlins/diamond.gif[/img]. Button folds, blinds fold. UTG+1 calls, Villain Calls.

(10.5 SB, 3 players) Flop: 3[img]/images/graemlins/diamond.gif[/img]3[img]/images/graemlins/club.gif[/img]9[img]/images/graemlins/spade.gif[/img]
UTG+1 checks, Villain bets, Hero raises, UTG+1 folds, Villain calls.

(7.25 BB, 2 players) Turn: 3[img]/images/graemlins/diamond.gif[/img]3[img]/images/graemlins/club.gif[/img]9[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/diamond.gif[/img]
Villain checks, Hero bets, Villain raises.

There are 10.25 big bets in the pot, and it will cost me 2 BB to see a showdown in a 13.25 pot, so I'm getting about 5.6:1 to see a showdown with my AK. My AK has to be good about 15% of the time here to make this calldown. I'm not sure whether it is or whether it's not.

But what also bothers me here are the odds that the villain is getting on his check-raise, no matter what he's holding. There are 8.25 BB in the pot when I bet the turn, so when he invests 2 BB on his check-raise, I only have to fold about 19% of the time for his check-raise to be profitable. I fold to a random turn check-raise much more often than 19% of the time.

So, the turn check-raise scares me because, well, it's a turn check-raise, but I'm really only scared of a pair here. Unless he's got 99, I think villain is relatively unlikely to have hit this 3392 board given his action so far, but the check-raise is a strong move with me holding overcards and villain could easily have AA-88. Maybe he's picked up a flush draw. Maybe he's got nothing. I think I'm probably beaten, but at the same time I don't like folding in this biggish pot.

My thinking is all screwed up here, because it seems like I have no good move: If I call down, I feel like I don't have the odds and it is -EV and calling down with overcards is a leak that I've tried to plug. But the pot is pretty big and so I feel like folding is also -EV.

Could someone help me straighten out my thinking here? Is the bet bad because it gets me into this spot? Should I be check-raising turns on rag boards after I get 3-bet preflop? Responses like "fold" or "call down" are appreciated, but I'm really trying to learn how to think about situations like this.