With the flush hitting on the river I check behind and thank whoever is up above that I split. If I didn't think he had the nut straight on the turn then I'm seriously considering him drawing to the flush (but I'm kinda scared like that).
A braver man than me would shove the river, representing the flush. Don't see many opponents raising your river bluff with less than the nut flush so you might well get away with a half pot bluff on the river. If you have any degree of confidence that he's not playing the flush draw here then a half pot river bet figures to make a profit in the long run. If I half pot the river, I fold to a raise.
Me. I'd just check behind.