We introduce an efficient technique for computing Casimird energies and forces between ob jects of arbitrarily complex 3D geometries. In contrast to other recently developed methods, our technique easily handles non-spheroidal, non-axisymmetric ob jects and ob jects with sharp corners. Using our new technique, we obtain the first predictions of Casimir interactions in a number of experimentally relevant geometries, including crossed cylinders and tetrahedral nanoparticles.