I am really interested to see the answers to this question - because my initial instinct is that you played this hand more or less correctly - but I have a nagging feeling that I am wrong.

The difficulty for me is that I can't see how the 3 made his straight, since that would mean he saw 3 bets (2 pre-flop and 1 flop) with a very weak holding.

He may have pocket 3s, but that is also pretty weak to call with that flop.

In short, in order to have us beat, I am thinking that he would have had to make some pretty loose calls in the first two rounds of betting.

I am thinking that perhaps he has something like T9 of clubs - so the 3 gave him the flush draw, significantly strengthening his hand.