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, ~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.