We present a tunable plasmon-induced transparency (PIT) structure that is composed of dielectric grating and a graphene system to manipulate terahertz (THz) waves. The graphene system consists of a graphene sheet and a graphene ribbon layer, with a spacer between them. By exploiting the diffraction coupling of THz wave with dielectric grating, graphene plasmonic resonance is efficiently excited on both graphene sheet and graphene ribbons. This leads to the surface plasmon mode of the graphene sheet and the localized plasmon mode of the graphene ribbons. The coupling between the two-plasmon modes via near-field destructive interference generates a strong PIT effect with slowing the group velocity of THz waves. A group delay over 0.2 ps and group index beyond 170 can be achievable. The group slowing effect is dynamically tunable with varying the Fermi level of graphene. The work suggests a promising scheme for on-chip graphene slow-wave devices at the THz regime.