Internal degrees of freedom and periodic structure are critical requirements in the design of acoustic/elastic metamaterials since they can give rise to extraordinary properties like negative effective mass and stiffness. However, they are challenging to realize in three dimensions. Closed-cell, crystalline foams are a particularly advantageous basis to develop metamaterials as they intrinsically have a complex microstructure, exhibiting internal resonances. Recently self-assembly techniques have been implemented to produce such foams: a Kelvin (body centered cubic) foam, a face centered cubic foam, and a Weaire-Phelan structure. Numerical models are employed to demonstrate that such foams are superanisotropic, selectively behaving as a fluid or a solid, pentamode solids as a result of fluid-structure interaction, in addition to having regimes characterized by film resonances and high density of states. Microstructural deformations obtained from numerical models allow the derivation of equivalent mechanical models. (C) 2014 Acoustical Society of America.