We study the early stages of spinodal decomposition in binary liquid crystal mixtures by solving linearized time-dependent Landau-Ginzburg equations for concentration (conserved order parameter) and two orientation (nonconserved) order parameters, one for each component. The theory takes into account a cross term between concentration and orientation gradients, which becomes an important factor for phase separation kinetics. We calculate growth rates and structure factors for the order parameters, following thermal quenches from a stable isotropic state into an unstable region. Depending on the quenched temperature and concentration, we find five unstable regions which are characterized by the time evolutions of the structure factors.