Preflop
UTG: raise 3$
MP Hero: K K re-raise 12$
all other FOLD
UTG: call

FLOP AT9
UTG: check,
Hero: bet 16,5$,
UTG: call

Turn
3
UTG: check
Hero: check

River
K
UTG: bet 22,5$
Hero: ?

The player at UTG was quite tight. He played 3 hands in an hour of play. He never made it to the showdown, so I had no read on him. I'm not sure if I played this game correctly. What do you think?

Check this flop. It's a really bad flop for KK, especially KK without a diamond. In general you want to check monotone flops or bet them small. You almost never want to bet big and it certainly wouldn't be with a hand like yours.

There are quite a few hands with a single diamond that can call the flop. The river being the Kd is good in one sense in that there are very few flopped flushes he can have (apart from the royal, they are all smaller suited connectors, and do they call preflop?) however UTG would probably call your flop bet with AQd, AJd (if this is called preflop), QdQ and JdJ. That's quite a few hands.

You only need to be good slightly more than 20% of the time so calling probably isn't a mistake, there are enough hands (TT, 99, some bluffs although I'm struggling to find them - they would probably be Ax) to get close to that ratio but folding can't be bad either. But concentrate on the flop which is more important than this decision.

Remember to put effective stack sizes in at the start of your post.

Prefer to check flop.
Delayed small c-bet turn.

Kd is nut worst river diamond for H. V has way more QdX. H would have bet AxQd on turn.

Even at 7-2, this is a fold to a tight V.

If villain is capable, I would call. If he isn't fold. There could be some busted straight draws and primarily AQo without a diamond thst decides to turn their hand into a bluff. Once in a blue moon a lower set will bet for value. Your hand blocks AK that primarily seems like it would check river, but it doesn't block bluffs. Your hand is better to call with than AA which blocks bluffs

What are the effective stack sizes preflop? $12 seems too big of a raise in position. 9-11 seems more reasonable.

On monotone flops we shouldn't be betting so large. This is like a 20% pot spot with anything that wants to bet, and your hand mainly wants to check.