We present the development and validation of an efficient numerical wave tank (NWT) solving fully nonlinear potential flow (FNPF) equations. This approach is based on a variation of the 3D-MII (mid-interval interpolation) boundary element method (BEM), with mixed Eulerian-Lagrangian (MEL) explicit time integration, of Grilli et al., which has been successful at modeling many phenomena, including landslide-generated tsunami, rogue waves, and the initiation of wave breaking over slopes. The MEL time integration is based on a second-order Taylor series expansion, requiring to compute high order time and space derivatives. In order to solve wave-structure interaction problems with complex geometries, we reformulate the model to use a 3D unstructured triangular mesh, building on earlier work, but presently only working with linear elements. The added flexibility of arbitrary meshes is demonstrated by modeling the longitudinal forces on a truncated (surface-piercing) vertical cylinder, comparing to theory and experiment. In order to improve the computational efficiency of the BEM, we apply the fast multipole method (FMM), in the context of the new unstructured mesh. A detailed study of the resulting computational time shows both the efficiency of the earlier 3D-MII approach and the proposed one, and also what is necessary to scale such results up to larger grids.