New {TbCu3} and {DyCu3} single-molecule magnets (SMMs) containing a low-symmetry Ln(III) center (shape measurements relative to a trigonal dodecahedron and biaugmented trigonal prism are 2.2-2.3) surrounded by three Cu-II metalloligands are reported. SMM behavior is confirmed by frequency-dependent out-of-phase ac susceptibility signals and single-crystal temperature and sweep rate dependent hysteresis loops. The ferromagnetic exchange interactions between the central Ln(III) ion and the three Cu-II ions could be accurately measured by inelastic neutron scattering (INS) spectroscopy and modeled effectively. The excitations observed by INS correspond to flipping of Cu-II spins and appear at energies similar to the thermodynamic barrier for relaxation of the magnetization, similar to 15-20 K, and are thus at the origin of the SMM behavior. The magnetic quantum number M-tot of the cluster ground state of {DyCu3} is an integer, whereas it is a half-integer for {TbCu3}, which explains their vastly different quantum tunneling of the magnetization behavior despite similar energy barriers.