A dualband terahertz (THz) absorber including periodically distributed cross-shaped graphene arrays and a gold layer spaced by a thin dielectric layer is investigated. Numerical results reveal that the THz absorber displays two perfect absorption peaks. To elucidate the resonant behavior, the LC model is introduced to fit the spectra. Moreover, the strength and linewidth of the absorption peak can be effectively tuned with structural parameters and the relaxation time of graphene. Owing to its rotational symmetry, this THz absorber exhibits polarization insensitivity. Our designed absorber is a promising candidate in applications of tunable optical sensors and optical filters.