Jessee and I were in the snorkeling group. I preferred that because I really need the exercise these days. Basically we just swam around until somebody spotted one and then we all gave chase, using the boat to ferry snorkelers ahead. It's works sort of like a relay race until the turtle is so tired it can't run anymore. At that point it usually sits on the bottom and we're able to dive down and grab it.
The snorkeling group managed to get two and the Manta Tow group got one - two Hawksbills and a Green (it's a shame we weren't after Eagle Rays because there were loads of them). I made my first capture using this method, though the credit is mostly due to John who is incredibly good at swimming after and wearing the turtles out. He let me get one of them anyway. I dove down and grabbed on on the first try, but I also got hold of a clump of soft corals. I let go with that hand to release the coral and the turtle wiggled loose. Fortunately it was very tired and didn't make it far before settling on the bottom again. This time it cleverly nestled into a patch of fire coral. I dove down again, but reached for it cautiously so as not to be stung by the coral, and it got away again! It wasn't until my fourth attempt that I managed to hold on and bring it to the surface.
It was a lot of fun to get out and spend the morning in the water. Jessee really enjoyed it too - she was commenting how she rarely has time to get out there anymore. It makes a big difference to have something out of the routine.
(Don't worry, putting them on their backs doesn't hurt them! It actually calms them down and keeps them from crawling around and getting into trouble)