The scattering of electromagnetic waves by periodic magnetodielectric structures with arbitrary profiles and inhomogeneous distribution of permittivity and permeability over a period is analyzed using rigorous integral equations of macroscopic electrodynamics and partitioning a structure into thin layers. The solution is found for E-polarized and H-polarized plane waves. Numerous examples of the problem solutions are compared with previously published results. Examples of approximation perfectly conducting periodic structures by magnetodielectric structures with the same geometry of a period are given. For illustration of the presented method effectiveness the analysis applies to a grating composed of circular magnetodielectric coaxial cylinders.