A few clarifications:
Unidirectional carbon-fiber reinforced epoxy is just as stiff in compression as it is tension, but it is not as strong. The reason is that, under stress, the fibers undergo out-of-plane buckling that limits the strength of the material to about that of the resin alone. Unidirectional composites placed in flexure, where the neutral bending axis falls somewhere in the middle of the beam, will have properties in between the pure compressive and tensile properties. Overall, properties are highest in pure tension.
Howard is correct that the stiffest neck will result from having the CF beam nearest the player's thumb because that places the carbon in line with the highest tensile stresses inside the neck.
As mentioned, there are many grades of carbon fiber, including both short and continuous fibers. Carbon fiber is generally a "graphitic" structure that is analogous to a rolled-up newspaper. However, there are differences in the orientation and defects of the graphitic structure. In the best case, you have single-crystal graphite "whiskers." These will generally not be found outside of specialized (and high-cost) aerospace applications. It is technically a misnomer to refer to CF as "graphite." As Rick mentions, there are also carbon nanotubes, which can be though of as the smallest unit of a graphite whisker.
Resin matrix stiffness could be enhanced with carbon nanotubes to reduce damping in CF-reinforced beams.... However, organoclay is a much cheaper and more readily available alternative. In fact, epoxy will bond with untreated montmorillonite, which is dirt cheap because it is literally dirt.
-Ben
