We calculate the spin-wave spectra of two-dimensional composite materials consisting of periodic square arrays of parallel cylinders made of a ferromagnetic material embedded in a ferromagnetic background. Each material is described by its spontaneous magnetization MS and exchange constant A. An external static magnetic field is applied along the direction of the cylinders and both ferromagnetic materials are assumed to be magnetized parallel to this magnetic field. We consider the spin-waves propagation in the plane perpendicular to the cylinders. We reveal the existence of gaps in the magnon band structure of composite systems such as the periodic array of Fe cylinders in an EuO matrix. We investigate the existence of these gaps in relation to the physical parameters of the materials involved. We also study the influence of the lattice parameter (i.e., the square array periodicity) and the effect of the filling fraction of the cylinders on the magnon band structure.