In the current work, we elaborate two-dimensional metamaterials of controlled anisotropy. To that scope, we employ diamond and octagon-shaped planar lattices with and without inner links. Using a dedicated homogenization technique, we derive closed-form expressions for the lattice’s effective mechanical properties. We analyse the effect of the lattice’s configuration on the metamaterial’s effective static properties, identifying configurations with mechanical attributes desirable for morphing, biomedical and mechanical engineering applications. We thereafter compute the lattice’s wave propagation characteristics, deriving a link between the metamaterials’ static and dynamic properties. In particular, we analyse the longitudinal and shear wave phase velocity dependence on the lattice’s geometric configuration. Thereupon, we identify architectural arrangements for which the phase velocity vanishes in certain propagation directions, exhibiting wave propagation isolation characteristics. We demonstrate that the detected isolation features can systematically arise for lattice architectural designs that yield highly anisotropic static properties (thus high material moduli ratios) and anti-auxetic material behaviours (thus non-negative Poisson’s ratio values).