Good question. I remember wondering the same thing during my plant physio classes! I think about orchids way too much.
I think orchids are mainly C3 and maybe CAM. I looked it up right now, and it pretty much confirmed what I thought, though I was surprised that there may be some C4 orchids as well.
If you can, try to get a hold of "The physiology of tropical orchids in relation to the industry" (2004). It has some good stuff about carbon fixation in orchids. Or go to Google books, most (if not all) of the book is viewable. Really interesting book, I think I'll go see tomorrow if the university library has it. It's very informative reading!
From what I read of it now, generally (but like always, there are exceptions!) it seems that thin leaved orchids fall into C3 and thick leaved orchids have all the characteristic of CAM fixation.
There is some evidence that some Cyms and a handful of other genera may be C4, but there is little concrete evidence for now.
What are you studying that involves learning carbon fixation? I love meeting other will be plant scientists here!