The fragment-based hybrid many-body interaction (HMBI) model provides a computationally affordable means of applying electronic structure wavefunction methods to molecular crystals. It combines a quantum mechanical treatment of individual molecules in the unit cell and their short-range pairwise interactions with a polarizable molecular mechanics force-field treatment of long-range and many-body interactions. Here, we report the implementation of analytic nuclear gradients for the periodic model to enable full relaxation of both the atomic positions and crystal lattice parameters. Using a set of five, chemically diverse molecular crystals, we compare the quality of the HMBI MP2/aug-cc-pVDZ-level structures with those obtained from dispersion-corrected periodic density functional theory, B3LYP-D*, and from the Amoeba polarizable force field. The MP2-level structures largely agree with the experimental lattice parameters to within 2%, and the root-mean-square deviations in the atomic coordinates are less than 0.2 Å. These MP2 structures are almost as good as those predicted from periodic B3LYP-D*/TZP and are significantly better than those obtained with B3LYP-D*/6-31G(d,p) or with the Amoeba force field.