We report on the eigenmodes of photonic crystals consisting of submicron homogeneous chiral spheres in a nonchiral isotropic medium, by means of full electrodynamic calculations using the layer-multiple-scattering method. It is shown that resonant modes of the individual spheres give rise to narrow bands that hybridize with the extended bands of the appropriate symmetry associated with light propagation in an underlying effective chiral medium. The resulting photonic dispersion diagram exhibits remarkable features, such as strong band bending away from the Bragg points with consequent negative-slope dispersion inside the first Brillouin zone and sizable frequency gaps specific to each polarization mode. We present a rigorous group-theory analysis to explain features of the calculated photonic band structure, peculiar to a system which possesses time-reversal but not space-inversion symmetry, and discuss some interesting aspects of the underlying physics.