We compute the stimulated Brillouin scattering (SBS) gain for a metamaterial comprising a cubic lattice of dielectric spheres suspended in a background dielectric material. Theoretical methods are presented to calculate the optical, acoustic, and optoacoustic parameters that describe the SBS properties of the material at long wavelengths. Using the electromagnetic and strain energy densities, we accurately characterize the optical and acoustic properties of the metamaterial. From a combination of energy density methods and perturbation theory, we recover the appropriate terms of the photoelastic tensor for the metamaterial. We demonstrate that electrostriction is not necessarily the dominant mechanism in the enhancement and suppression of the SBS gain coefficient in a metamaterial, and that other parameters, such as the Brillouin linewidth, can dominate instead. Examples are presented that exhibit an order of magnitude enhancement in the SBS gain as well as perfect suppression.