TPTK facing 4-flush board

[PennDisc]

These situations are so read dependent - but what do you do facing an unknown? I only played 1 orbit when this hand came up. Villain hadn't played a hand yet, so he's tight, and I have seen nothing of his postflop play.

Party Poker 10/20 Hold'em (10 handed) converter

Preflop: Hero is UTG with A[img]/images/graemlins/diamond.gif[/img], K[img]/images/graemlins/diamond.gif[/img].
<font color="#CC3333">Hero raises</font>, <font color="#666666">3 folds</font>, <font color="#CC3333">MP2 3-bets</font>, <font color="#666666">5 folds</font>, Hero calls.

Flop: (7.50 SB) A[img]/images/graemlins/heart.gif[/img], 3[img]/images/graemlins/heart.gif[/img], 4[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
Hero checks, <font color="#CC3333">MP2 bets</font>, <font color="#CC3333">Hero raises</font>, MP2 calls.

Turn: (5.75 BB) 7[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets</font>, <font color="#CC3333">MP2 raises</font>, Hero ???

I'm not worried about giving free cards anymore after the turn card hits because he's either got the flush or he doesn't, so should I be check-calling from there? Bet-fold? Basically, do I need to get to showdown in a headsup situation like this or can I give up if he gives me indication that I'm beat?