I found myself in a similar situation with, ironically enough, quad 6's. I was in a little better situation though because I had pocket 6's and flopped the other 2. I was first to act as well and decided to just check and hope that someone was going to do the betting for me. Luckily, someone holding Big Slick decided to lead out on the betting and was followed by someone with a pocket pair and someone else holding something I can't readily recall. On the turn I check-raised and got 2 callers and then I bet out on the river.

More important than what YOU should do in those instances is hoping that someone else has made their 2nd best hand and will help you out by doing your betting for you until it's time to take over the betting. More often than not though when you have this kind of situation most people will fold because they have not made their hand.