The rate of exciton spin relaxation (flips) between the bright exciton states (F = +/-1) of CdSe nanocrystals is reported as a function of shape, for dots and nanorods. The spin relaxation is measured using an ultrafast transient grating method with a crossed linearly polarization sequence. It is found that the spin relaxation rate depends on the radius, not length, of the nanocrystals. That observation is explained by deriving an expression for the electronic coupling matrix element that mixes the bright exciton states.