We examine the linear elasticity of the bcc and gyroid phases of block copolymer melts with self-consistent field theory (SCFT). Linear elastic moduli for single crystals are predicted by calculating the free energies for slightly deformed crystal structures. Predicted Voight and Reuss bounds for the shear modulus of a polycrystalline material are quantitatively compared to the cubic plateau moduli found in linear viscoelastic measurements of both phases with good agreement. We also consider a model of pairwise additive interactions between "micelles" in the bcc and fee phases and find that it predicts ratios of elastic constants consistent with those predicted by SCFT for the bcc phase, but not for the fcc phase.