We investigate the effect of finite thickness of quantum wires on spin-density excitations (SDEs). In contrast to plasmon excitations, SDEs depend sensitively on the force law that describes the electron-electron interaction. We find that, especially for low electron densities, the corrections due to a three-dimensional interaction are very important. We use both the time-dependent Hartree-Fock and local-density approximations to calculate the first confined collective SDE. The results are compared with Raman experiments. The agreement is excellent and shows that a three-dimensional treatment is absolutely necessary to explain the experiments.