When an applied magnetic field has an arbitrary direction with respect to the lattice axes of a periodically microstructured or nanostructured metamaterial, the effective permittivity tensor of the metamaterial sample becomes anisotropic and all its components can be nonzero. This is true even if the microstructure has a high symmetry, e.g., cubic or triangular. It is found that the strong magneto-induced anisotropy which appears in the macroscopic response leads to an unusually strong anisotropic behavior of the Voigt effect and other magneto-optical effects.