Recent developments in directed photocuring of polymers have enabled fabrication of periodic lattice structures with highly tailorable geometries. The present study addresses the mechanics of compressive deformation of such structures with emphasis on the effects of strut slenderness , strut inclination angle , and number of repeat lattice layers . We present analytic models and finite element calculations for a broad parameter space and identify designs that yield desirable combinations of specific strength and energy absorption. The optimal designs (those for which crushing occurs at nearly constant compressive stress) are found to be those in which there is only one pyramidal layer, the inclination angle is of intermediate value ( = 50 deg) and the strut slenderness ratio falls below a critical value, typically . The performance of near-optimal structures is attributable to the balance between two competing processes during plastic deformation: (i) geometric hardening associated with lateral expansion of the nodes and the struts, and (ii) geometric softening arising from the corresponding reduction in strut angle. Comparisons with stochastic foams show that the lattice structures can be designed to attain levels of energy absorption not possible by foams (by factors of 3–5 on a mass basis), albeit at higher stress levels than those required for crushing foams.