Architected materials with superior energy absorption properties when loaded in compression. In several embodiments such materials are formed from micro-truss structures composed of interpenetrating tubes in a volume between a first surface and a second surface. The stress-strain response of these structures, for compressive loads applied to the two surfaces, is tailored by arranging for some but not all of the tubes to extend to both surfaces, adjusting the number of layers of repeated unit cells in the structure, arranging for the nodes to be offset from alignment along lines normal to the surfaces, or including multiple interlocking micro-truss structures.