You should have bet the flop. You usually have the best hand and it makes the rest of the hand easier to play.

Why did you check this flop? How did you expect him to react on the turn?

You have shown weakness and he could very well could be betting the turn with nothing, a draw, or Ace high. A check on the flop and a raise on the turn would look like a slowplay and you could force out a T or pocket 9's. That being said, I am confident that a flop bet is the standard play here.

I think that the flop-check, turn-call, river-fold is one of the worst ways to play this situation. If he is bluffing the turn, many players will still fire away at the river a significant portion of the time.