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Periodic lattices offer enhanced mechanical and dynamic properties per unit mass, and the ability to engineer the material response by optimizing the unit cell. Characterizing the effective properties of these lattice materials through experiments can be a time consuming and costly process, so analytical and numerical methods are crucial. Specifically, the Bloch-wave homogenization approach allows one to characterize the effective static properties of the lattice while simultaneously analyzing wave propagation properties such as band gaps, propagating modes, and wave directionality. While this analysis has been used for some time, a thorough study of this approach on three-dimensional (3D) lattice materials with different symmetries and geometries is presented here. Bloch-wave homogenization is applied to extract the effective stiffness tensor of 3D periodic lattices and confirmed with elastostatic homogenization, both within a finite element framework. Multiple periodic lattices with cubic, transversely isotropic, and tetragonal symmetry, including an auxetic geometry, over a wide range of relative densities are analyzed. Further, this approach is used to analyze 3D periodic composite structures, and a way to tailor their overall anisotropy is demonstrated. This work can serve as the basis for nondestructive evaluation of metamaterials properties using ultrasonic velocity measurements. [ABSTRACT FROM AUTHOR]