Seeing as that you are UTG, I would definitely bet for 2 reasons:

1. By the looks of it, you most likely have the pot with KK and a decent kicker after the flop. You haven't seen a raise representing the K or 2 pair. If I had a king or 2 pair, I would have raised you for sure (to see where I was at, and ditch the chasers)!!

2. I'd also bet to push out the possibility of somebody betting into me. If you check and receive a large bet, I imagine you would fold out of fear. If you stayed in, you would have to commit more chips when there is a possibility you may be beat. Bet first and watch the reactions!

Therefore, I believe that betting is the only option.... wait one more reason I would bet:

3. If I was your opponent and noticed your hesitance and contemplation, I would bet into you regardless of what I had. Don't give your opponent the chance to do this. Push them out.

Let me play out a quick scenario...

Opponent is holding K8 suited,
"Flop comes K 4 8 rainbow"
I might check raise in early position, but bet in late position. I want to take the pot down and get rid of the chasers now. It is already a decent size pot with 3 bets and callers. I wouldn't just call with this if I was bet into. I would demand more chips from you.

Also, why would anybody holding something other than K, 4 or an 8 continue with this pot. Possibly to bluff?? The only hands you fear are AK KQ K4 K8 (preflop) K9 (turn). With no raises, no straight or flush chances, you have to bet.

(side note, if you are under the gun, how come there are 2 checks on the turn beofre your bet??? doesn't add up. if you are UTG, BB checks, then your turn to act then LP.... correct??)