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We consider a host of regular lattice architectures and present analytical and computational approaches to derive the structure-property relationship for such structured material by exploiting the translational periodicity of infinite lattices. Two specific geometries - the so-called hexagonal honeycomb and the woodpile lattice - are studied analytically and computationally. The elasto-plastic response in the case of the first, and the bending response of lattice beams for the second, is considered. These specific problems have been motivated by biostructures relevant to medical implants and scaffolds. We also present novel methods to additively manufacture such lattices. When possible, the response is obtained as an analytical function of the microstructural parameters described by the geometry of the repetitive elements of the lattice, such as characteristic diameter, length, or thickness. Alternative methods of manufacturing materials with random internal architectures are also presented. The relative strengths and weaknesses of the two classes of materials with respect to analysis and manufacture are discussed. [ABSTRACT FROM AUTHOR]