  The magnetic structure of LaMnO$_3$ is investigated on three-dimensional
clusters of MnO$_6$ octahedra by using a combination of relaxation and Monte
Carlo techniques. It is found that the cooperative Jahn-Teller phonons lead to
the stabilization of A-type antiferromagnetic and C-type orbital structures in
the physically relevant region of parameter space for LaMnO$_3$ with small
corrections due to tilting effects. The results suggest that strong Coulomb
interactions are not necessary for a qualitative description of undoped
manganites. In fact, it is shown that the present result is not essentially
changed even if the Coulomb interaction is explicitly included.
