A calculation of the optical band structure of a three dimensional lattice of resonant two-level atoms in the dipole approximation is presented. The formation of band gaps is exhibited and confirmed by a calculation of the density of states. The band structure can be characterized by two dimensionless parameters. We find a longitudinal polarization mode as well as a class of vacuum modes that are unaltered by the interaction with matter. Numerical calculations are performed for a face centered cubic lattice; other lattices can be evaluated as easily.