We derive the Ewald representation for the dyadic periodic Green's functions to represent the electromagnetic field in a three dimensional (3D) periodic array of electric and magnetic dipoles. Then we use the developed theory to analyze the modes with real and complex wave number in a 3D periodic lattice of lead telluride (PbTe) microspheres at infrared frequencies and in a 3D periodic lattice of titanium dioxide (TiO2) microspheres at millimeter waves. Each microsphere is equivalently modeled with both an electric and a magnetic dipole, via a method here called the dual dipole approximation (DDA). The 3D lattices exhibit first a magnetic-induced then an electric-induced feature determined by microsphere magnetic and electric resonances. The DDA wave number results are compared to the ones computed with single electric or single magnetic dipole approximation and to the ones retrieved by using the Nicolson-Ross-Weir (NRW) retrieval method from reflection and transmission of finite thickness slabs computed by a full-wave simulation. It is shown that the DDA method is in very good agreement with NRW, in contrast to the previously reported single dipole approximation methods that fail to predict one of the two features (either electric or magnetic). A mode with transverse polarization is found to be dominant and able to propagate inside the lattice, and therefore the composite material can be treated as a homogeneous one with effective refractive index. This is obtained by adopting five different retrieval procedures for each lattice, and their agreement or disagreement is discussed.