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Based on Shore and Yaghjian's work (IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3077-3091, Oct. 2009), a general theory has been developed to describe traveling waves on three-dimensional (3-D) periodic arrays of two sets of magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice. This theory is eventually in the form of k-β (dispersion) equations. To improve the computational efficiency, rapidly converging expressions and their double summation form are derived for slowly converging summations in the k-β equations. The dispersion diagrams of seven different arrangements of the spheres are analyzed for three combinations of sphere types: 1) dielectric spheres with equal permittivity but different radius; 2) dielectric spheres with equal radius but different permittivity; and 3) one set of spheres is purely dielectric while the other set is magnetic. Results show that the maximum bandwidths of the double-negative (DNG) region provided by different spheres arrangements for spheres combinations 1-3 are, respectively, 0.21%, 0.069%, and 7.403%. Compared to results reported in previous literature, analysis of these possible arrangements of the spheres shows similar narrow DNG bandwidths for spheres combinations 1 and 2, and wider DNG bandwidths for spheres combination 3. Although purely dielectric materials with relative permittivity much greater than one are readily available, the usefulness of purely dielectric DNG metamaterials still depends on whether the narrow bandwidths achievable are acceptable for the particular applications. Since purely magnetic materials with relative permeability much greater than one above 1 GHz are not currently available, the practicality of fabricating DNG metamaterials using arrays with spheres combination 3 is questionable for radio frequency (RF) applications, at least at present, despite the fact that this combination yields much wider DNG bandwidths than those of spheres c- mbinations 1 and 2.