Methods of making truss-based periodic cellular solids that have improved structural properties and multifunctional design. Many materials (metals, ceramics, glasses, polymers, composites and even semiconductors) can be shaped into cellular, truss-like architectures with open, closed or mixed types of porosity and then very uniformly arranged in controlled, three-dimensional space-filling arrays. The truss-like elements do not necessarily have a constant cross-section, nor are they necessarily straight or solid throughout (they could be hollow). Their cross sections can be circular, square, triangular, I-beam or other shapes of interest depending on multifunctional needs. When bonded together by solid state, liquid phase, pressing or other methods at points of contact, a cellular structure of highly repeatable cell geometry and few imperfections results. The bonds hold the truss elements together in a desired configuration, allow load to be efficiently transferred amongst them and make the resulting structure significantly more rigid when bent, compressed or sheared. These constructed cellular solids offer a broad range of multifunctional structural uses with a tremendous freedom for choosing the truss type, orientation and distribution. Multiple materials can be intermixed.