We develop an approach to homogenize three-dimensional periodic solid-solid elastic composites with cubic lattice at low frequencies, by using plane wave expansion and perturbation theory with respect to the long wavelength limit. Based on the fact that the two shear waves propagating along lattice axis are degenerated, we derive formulae for effective velocities parallel and normal to the lattice axis, from which three independent effective elastic moduli axe calculated, respectively. Theoretical results, which take into account the multiple scattering and the structure of the periodic medium, are in good agreement with the previous isotropic theory at high-symmetry directions.