I play it exactly the same because hes not folding a flush, but thats not what I put him on anyway. Most people raise the flop with a hand they want to protect from a 4 card flush. If he has a set, hes got good odds to chase his boat unless I really blast him.
If I can make note of a minority opinion here in answer to rode dog's very good question:
I'd play the turn much harder with AdXd here, definitely betting out full pot. For one thing, the little flush still can't get away from his hand but can if a final diamond falls on the river. For another, I don't want to have to deal with a paired board on the river. If you get really lucky, 9d4d might move in on the turn to a full pot bet.
Yes, but a set is probably going to bet the turn if you check to him....but if YOU bet, he might get suspicious and just call. I wanted to get a set all in which is why I check raised all in.