I'd check behind on the turn, because it's too likely that BB called the flop bet with a high heart of his own, and I don't feel like paying two bets for it. If he doesn't have a heart, he's unlikely to improve to beat our hand, and he'll probably fold. There's a decent chance he'd also fold if he has a very low heart, though this is read-dependent.
It doesn't look like he's already got two pair or trips, so it's very unlikely the river card will change the outcome of this hand. One of you is drawing dead. Since you hold the T[img]/images/graemlins/heart.gif[/img], it's hard for you to tell we're you're at. If villian has A[img]/images/graemlins/heart.gif[/img], Q[img]/images/graemlins/heart.gif[/img], 6[img]/images/graemlins/heart.gif[/img], 5[img]/images/graemlins/heart.gif[/img], 4[img]/images/graemlins/heart.gif[/img], or 3[img]/images/graemlins/heart.gif[/img], he probably has a good idea of where he stands. If he's got J[img]/images/graemlins/heart.gif[/img] or 9[img]/images/graemlins/heart.gif[/img], he's probably less confident. Basically, he probably has more information than you, which is bad for you.
If you have him beat and check-behind on the turn, he may assume that you don't have a heart yourself. In that case, he may bet out or check-and-call, which earns you one more bet from him in the cases where he would have folded on the turn.
In summary, I like checking behind on the turn because:
<ul type="square">[*]It saves us two bets when we're way behind[*]It may earn us an extra bet when we're way ahead[/list]Unless you have a read that says that villian will call all the way down with junk even when it's really obvious that you have a flush. Then I'd say bet.