Bet the turn and definitely check-call this river, since the heart draw missed, as did the spade redraw. The only reason to not bet the turn is because you fear being semi-bluff raised on this turn (with two draws on the board and a pseudo-scary paired board), but this opponent may or may not be tricky enough, or even if he is, he may fear that you will three-bet him. So on balance, bet the turn.