10/20 Fired two barrels and the flush comes.. now what?

[FourKing Hell]

The enemy in this hand is 55/5 over 800 or so hands.

Party Poker No-Limit Hold'em, $ BB (10 handed) converter

Hero ($3209.5)
MP1 ($2419)

Preflop: Hero is UTG+1 with 9[img]/images/graemlins/spade.gif[/img], 9[img]/images/graemlins/heart.gif[/img]. SB posts a blind of $10.
<font color="#666666">1 fold</font>, <font color="#CC3333">Hero raises to $80</font>, <font color="#666666">1 fold</font>, MP1 calls $80, <font color="#666666">6 folds</font>.

Flop: ($190) 8[img]/images/graemlins/spade.gif[/img], 6[img]/images/graemlins/club.gif[/img], 9[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets $190</font>, MP1 calls $190.

Turn: ($570) 4[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">Hero bets $570</font>, MP1 calls $570.

River: ($1710) 3[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
Hero ???

I've seen this guy make some pretty crazy calls, but usually they were "semi-crazy", i.e. AT or so when an Ace hit the board. I'm not sure if I can put him on top pair or something here. Theoretically, calling the bet on the turn should rule out the flush draw, but with this guy you never know.

I find myself in similar situations quite often - I don't mean having top set, but being up against a possible flush draw that nevertheless called a big turn bet.

What to do? Check-call? Push? 'shudder' check-fold?

And what if I have aces instead? Tens?