Back when I was associated with some racing people, we got a 'preview' of the various composite technologies being toyed with over in Stuttgart. Normal organic polymer matrix composites can't stand up to combustion temps, but there are a few ways around this problem. One is to use a super high-temp matrix. Carbon matrix (hence the term carbon/carbon) is the most widely recognized. They make it basically by partially burning out the matrix of an organic polymer matrix part. Of course this is done under very well-controlled conditions. It matters a lot just exactly what polymer matrix you start with, too. These details are still carfefully guarded secrets.
Another way around the problem is to shield the part so it never really gets that hot. Ceramics are one way. Also, several companies make a coating (I've used the one from Sterling) that has amazing heat shielding properties. Something like this could be put on the top of the piston and in and above the ring land(s) where it normally runs really hot. You would probably still need a really HP resin system like bismaleamide rather than epoxy, but it's still doable.
Once you get away from thermoset type resins, there are even more possibilities. Amoco plastics has several thermoform resins that can and have been used(with the right fiber reinforcements, of course) to make any engine part, including pistons. These have been around since the early 80's. They used metal shielding for the piston crowns on their test engines, IIRC. Of course this type of plastic needs to be melted and injected into the mold which implies a pretty big capital investment. Those German boys were playing with all of these options.
The last barrier now is only cost. Can any of these materials compete with aluminum on cost? I doubt it. So they will remain for now the toys of the race car crowd, NASA, aerospace and the military.
Jimbo