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A novel solution is presented for the problem of three-dimensional electromagnetic scattering of a time-harmonic plane wave from a doubly periodic infinite array of disjoint finite-size penetrable bodies. A set of fictitious doubly periodic and properly modulated patches of magnetic current of cross polarization is used to simulate the scattered field. A set of fictitious elemental lectric dipoles of cross polarization is used to simulate the field inside the penetrable bodies. The complex amplitudes of the two sets of fictitious sources are adjusted to render the tangential components of the electric and magnetic fields continuous at a selected set of points on the surface of any of the scatterers. The suggested solution procedure is simple to implement and is applicable to doubly periodic arrays composed of homogeneous bodies of smooth, but otherwise arbitrary, shape. The accuracy of the method has been demonstrated by means of several accuracy checks. It has also been shown that in the limiting case of widely spaced spherical scatterers the numerical solution agrees well with an approximate analytic solution.