The behavior of an anisotropic Heisenberg model on a semi-infinite cubic lattice is studied. We employ the Green's function formalism and the random-phase approximation in our analysis. We consider different possible configurations for the anisotropy parameters of the surface and bulk. We determine the profile of the magnetization and the renormalized magnon energy spectrum as a function of temperature and of the anisotropy parameters of the surface and bulk. We also find the complete phase diagram for the multicritical points which is in agreement with the real-space renormalization group calculations. The onset of surface ordering takes place when the surface magnons become more energetic than the bulk modes. In the limit where bulk and surface are both described by a pure Heisenberg model, there is no long-range surface magnetic order over the paramagnetic bulk, whatever the values of the exchange couplings are. We relate this result with the Mermin-Wagner theorem. We show that the slope of the surface magnetization is discontinuous at the bulk critical temperature and that the discontinuity increases when the model on the surface varies from Ising to Heisenberg. In the particular case when the exchange couplings become antiferromagnetic, we have found the canted-paramagnetic critical field at the surface which is larger than that of the bulk. At very low temperatures, the critical field at the surface does not follow a T 3a2 Bloch law, but instead decreases linearly with temperature.