Our line on the river is a pretty simple check/call a reasonable bet. Maybe you could make a case for folding, but that's pretty weak.
As far as the turn bet, our opponent's calling range on the flop is like:
Middle Pairs: 88, 99, 7x
Draws: Two hearts, 56
Worse Kings: KQ, KJ, ect.
Stuff that Crushes us: 74, 77, 44
On the turn, hearts get there, and I doubt 7x (or even KT/K9) could sustain another bet. So yeah, I pretty much like checking the turn, and trying to get to showdown cheaply at this point.