<p>Two-scale homogenization provides an effective tool for the design of microstructured materials. In particular, when considering periodic locally resonant metamaterials with soft inclusions in a stiff matrix, asymptotic homogenization allows for the effective prediction of band gaps in the long wave regime. When inclusions lack rotational symmetry, the resulting homogenized medium is characterized by anisotropic stiffness and anisotropic frequency-dependent mass density, leading to more complex wave propagation behavior. In this work, by means of numerical and analytical examples, we show how the equivalent homogenized formulation is able to correctly reproduce the dispersion and wave transmission properties of the heterogeneous anisotropic medium. The role of the anisotropic equivalent mass tensor to selectively polarize elastic waves and to achieve mode conversion is elucidated in particular.</p>