In this paper, the photonic bands of the body centered cubic and the hexagonal close packed structures consisting of spherical air atoms are calculated using vector plane wave method. The results show that complete band gaps (covering the entire Brillouin zone) open up for both structures. It is revealed that the maximum refractive index modulation coincides with the widest band gaps and the optimum volume fraction of the atoms is structure independent. The photonic bands of the face-centered-cubic (FCC) structure consisting of spherical anisotropic dielectric atoms are also calculated. The results show that a complete band gap opens up, which is quite different from the FCC structure consisting of spherical isotropic dielectric atoms.