Open-cell foams made from a wide array of base materials (e.g. metals, polymers, and ceramics) are desirable for a broad range of applications including shock absorption, thermal and acoustic insulation among others. This talk will focus on the propagation of elastic waves and energy flow characteristics of 3D lattice materials and open-cell foams made of both regular and random microstructures. Dispersive wave beaming behavior and preferential directions of energy flow are examined through Bloch wave analysis and finite element simulations on unit periodic cells. Dynamic characteristics of several regular lattices such as a simple-, framed-cubic and the octet lattice, are studied to examine the effect of average nodal connectivities for both stretching- and bending-dominated behavior. Significant wave beaming is observed for the simple-cubic and octet lattices in the low frequency regime while a complete bandgap is found only for the frame- cubic lattice. Periodic open-cell foam topologies are then generated with the Surface Evolver and convergence studies on band diagrams of different domain sizes indicate that a RVE consists of at least 512 cells. Wave directionality is investigated by extracting phase and group velocity plots. The effects of topological asymmetry and microstructural randomness are studied in detail and excellent agreement is found between the wave behavior of a random foam and that of a regular Kelvin supercell model in the low-frequency regime. In higher modes and frequencies however, and as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam models increases.