  We derive from first principles an accurate homogenized description of
periodic metamaterials made of magnetodielectric inclusions, highlighting and
overcoming relevant limitations of standard homogenization methods. We obtain
closed-form expressions for the effective constitutive parameters, pointing out
the relevance of inherent spatial dispersion effects, present even in the
long-wavelength limit. Our results clarify the limitations of quasi-static
homogenization models, restore the physical meaning of homogenized metamaterial
parameters and outline the reasons behind magnetoelectric coupling effects that
may arise also in the case of center-symmetric inclusions.
