We present a methodology for identifying constitutive responses of crushable, linear-softening materials that would reduce the severity of brain injury caused by head impact in a typical automobile or sports collision. It is based on analysis of accelerations imparted to a spherical mass (representative of the human head) upon impact at prescribed velocity onto a flat padded structure. The resulting acceleration–time histories are used to calculate the corresponding Head Injury Criterion (HIC): a weighted product of acceleration and impact duration that has been found to correlate with the severity of brain injury. In the best-case scenario, the HIC is reduced by a factor of 1.84 relative to that obtained for a system optimized with a perfectly plastic foam. The optimal combinations of yield stress and crushing strain are not unique; that is, the optimum can be achieved with a range of strengths and crushing strains. The present solutions are expected to find utility in guiding the design of new polymer lattice materials for use in impact protection systems.