The viscoelastic behavior of block copolymer melts that exhibit a cubic phase has been examined by oscillatory shear experiments. A low frequency plateau in the measured storage modulus that is absent in the disordered phase is found for both gyroid and body-centered cubic sphere phases of diblock and triblock copolymer melts. The magnitude of the apparent plateau modulus is found to be insensitive to strain amplitude for strains of 0.05%-5%, and so is believed to be characteristic of the true linear response. The value of the apparent terminal relaxation frequency, beneath which G″(ω) > G′(ω) is, however, sensitive to strain amplitude in the same range of strains and decreases steadily with decreasing strain, indicating that the terminal regime is highly sensitive to nonlinear effects. The presence of one or more entangled blocks is found to decrease the terminal frequency, and thus extend the range of linear behavior. Experimental results for the dependence of the plateau modulus and the unit cell size upon molecular volume yield effective power law exponents that are close, but not identical, to those predicted by self-consistent field theory.