The asymmetric transmission effect has attracted great interest due to its wide modern optical applications. In this Letter, we present the underlying theory, the design specifications, and the simulated demonstration of tunable dual-band asymmetric transmission for circularly polarized waves with a graphene planar chiral metasurface. The spectral position of the asymmetric peak is linearly dependent on the Fermi energy and can be controlled by changing the Fermi energy. The success of tunable dual-band asymmetric transmission can be attributed to the enantiomerically sensitive plasmonic excitations of the graphene metasurface. This work offers a further step in developing tunable asymmetric transmission of circularly polarized waves for applications in detectors and other polarization-sensitive electromagnetic devices.