Cells play an active role in the maintenance of mechanical homeostasis within tissues and their response to elastic forces is important for tissue engineering. We predict the collective response of an ensemble of contractile cells in a three-dimensional elastic medium to externally applied strain fields. Motivated by experiment, we model the cells as polarizable force dipoles that change their orientation in response to the local elastic strain. The analogy between the mechanical response of these systems and the dielectric response of polar molecules is used to calculate the elastic response function. We use this analogy to evaluate the average cell orientation, the mean polarization stress, and the effective elastic constants of the material, as a function of the cell concentration and matrix properties.