The structural relaxation of multilayer graphene is essential in describing the interesting electronic properties induced by intentional misalignment of successive layers, including the recently reported superconductivity in twisted bilayer graphene. This is difficult to accomplish without an accurate interatomic potential. Here, we present a new, registry-dependent Kolmogorov-Crespi-type interatomic potential to model interlayer interactions in multilayer graphene structures. It consists of two parts, representing attractive interaction due to dispersion and repulsive interaction due to anisotropic overlap of electronic orbitals. An important new feature is a dihedral-angle-dependent term that is added to the repulsive part to describe correctly several distinct stacking states that the original Kolmogorov-Crespi potential cannot distinguish. We refer to the new model as the dihedral-angle-corrected registry-dependent interlayer potential (DRIP). Computations for several test problems show that DRIP correctly reproduces the binding, sliding, and twisting energies and forces obtained from ab initio total-energy calculations based on density-functional theory. We use the new potential to study the structural properties of a twisted graphene bilayer and the exfoliation of graphene from graphite. Our potential is available through the OpenKIM interatomic potential repository at https://openkim org.