Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute-intensive is to use multiscale methods that solve coarsened flow problems by use of a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, the multiscale finite-volume (MsFV) method, is well-studied for 2D Cartesian grids but has not been implemented for stratigraphic and unstructured grids with faults in three dimensions. We present an open-source implementation of the MsFV method in three dimensions along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using generalized minimal residual (GMRES) method or as a preconditioner in combination with a standard inexpensive smoother. We conduct a series of numerical experiments in which approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeabilities and geometries. On the one hand, the results show that the MsFV method can produce accurate approximations for geological models with pinchouts, faults, and nonneighboring connections, but on the other hand, they also show that the method can fail quite spectacularly for highly heterogeneous and anisotropic systems in a way that cannot efficiently be mitigated by iterative approaches. Thus, the MsFV method is, in our opinion, not yet sufficiently robust to be applied as a black-box solver for models with industry-standard complexity. However, extending the method to realistic grids is an important step on the way toward a fast and accurate multiscale solution of large-scale reservoir models. In particular, our open-source implementation provides an efficient framework suitable for further experimentation with partitioning algorithms and MsFV variants.