The electronic band structure of a solid is a collection of allowed bands separated by forbidden bands, revealing the geometric symmetry of the crystal structures. Comprehensive knowledge of the band structure with band parameters explains intrinsic physical, chemical, and mechanical properties of the solid. Here we report the artificial polaritonic band structures of two-dimensional honeycomb lattices for microcavity exciton-polaritons using GaAs semiconductors in the wide-range detuning values, from cavity photonlike (red-detuned) to excitonlike (blue-detuned) regimes. In order to understand the experimental band structures and their band parameters, such as gap energies, bandwidths, hopping integrals, and density of states, we originally establish a polariton band theory within an augmented plane wave method with two-kind bosons, cavity photons trapped at the lattice sites, and freely moving excitons. In particular, this two-kind band theory is absolutely essential to elucidate the exciton effect in the band structures of blue-detuned exciton-polaritons, where the flattened excitonlike dispersion appears at larger in-plane momentum values captured in our experimental access window. We reach an excellent agreement between theory and experiments in all detuning values.