A mathematical model is developed for propagation of electromagnetic waves in periodic magnetic nanostructures. The model is based on the decomposition approach and autonomous blocks (ABs) with magnetic nanoinclusions and Floquet channels. The Floquet theorem is used to derive the characteristic equation for constants of wave propagation in 3D periodic nanolattices. The equation involves the scattering matrix (descriptor) of ABs. The propagation constants of longitudinal (right- and left-handed polarized) waves and transverse (ordinary and extraordinary) waves in a lattice of ferromagnetic metal nanospheres are calculated in the microwave band as functions of the lattice period. A technique is developed for electrodynamic calculation of the effective electromagnetic parameters of nanostructured gyromagnetic media, and the complex values of the components of the effective tensor permeability and permittivity are found as functions of the ratio of the nanoparticle radius and the lattice period.