This review summarizes progress about a recent homogenization theory based on the Fourier formalism for solid phononic crystals, which is valid for arbitrary Bravais lattice and any form of inclusions in the unit cell. The theory provides explicit expressions for the tensors of the effective nonlocal elastic response (dependence on frequency and wave vector), namely the effective dynamic mass-density and compliance matrices. With the use of this theory, our predictions in the quasistatic limit for one and two-dimensional phononic crystals coincide with those of finite-element and asymptotic-homogenization methods. It is also shown that the derived expressions can be applied to phononic crystals with liquid components (two-dimensional sonic crystals) and agree with predictions of the multiple scattering theory. The formalism of non-local effects is fully demonstrated only for a one-dimensional elastic metamaterial having simultaneously negative effective dynamic mass density and elastic shear modulus. The development and applications of this homogenization theory, unlike other formalisms, arises from the inspiration of intense research efforts to simultaneously describe local and non-local effective properties in elastic periodic structures.