The complex band structures of a 1D anisotropic graphene photonic crystal are investigated, and the dispersion relations are confirmed using the transfer matrix method and simulation of commercial software. It is found that the result of using effective medium theory can fit the derived dispersion curves in the low wave vector. Transmission, absorption, and reflection at oblique incident angles are studied for the structure, respectively. Omni-gaps exist for angles as high as 80° for two polarizations. Physical mechanisms of the tunable dispersion and transmission are explained by the permittivity of graphene and the effective permittivity of the multilayer structure.