We present a homogenization theory for elastic phononic crystals. The theory provides explicit formulas for the calculation of the effective tensors of the acoustic response, which turns out to be nonlocal in the most general case. We have compared the predictions of our theory for 1D and 2D phononic crystals with those of previous local approaches. In particular, we have applied the developed theory to 1D phononic crystals composed of auxetic materials. We found that such systems exhibit metamaterial behavior at frequencies corresponding to the second band of the phononic dispersion.