Dispersion properties of electromagnetic crystals formed by small uniaxial resonant scatterers (magnetic or electric) are studied using the local field approach. The goal of the study is to determine the conditions under which the homogenization of such crystals is possible. Therefore the consideration is limited to the frequency region where the wavelength in the host medium is larger than the lattice periods. It is demonstrated that, together with the known restriction for the homogenization related to the large values of the material parameters, there is an additional restriction related to their small absolute values. On the other hand, the homogenization becomes allowed in both cases of large and small material parameters for special directions of propagation. Two unusual effects inherent to the crystals under consideration are revealed: a flat isofrequency contour that allows subwavelength imaging using the canalization regime and birefringence of the extraordinary modes which can be used for beam splitting.