  We study three-dimensional dimerized S=1/2 Heisenberg antiferromagnets, using
quantum Monte Carlo simulations of systems with three different dimerization
patterns. We propose a way to relate the N\'eel temperature T_N to the
staggered moment m_s of the ground state. Mean-field arguments suggest that T_N
is proportional to m_s close to a quantum-critical point. We find an almost
perfect universality (including the prefactor) if T_N is normalized by a proper
lattice-scale energy. We show that the temperature T* at which the magnetic
susceptibility has a maximum is a good choise, i.e., T_N/T* versus m_s is a
universal function (also beyond the linear regime). These results are useful
for analyzing experiments on systems where the spin couplings are not known
precisely, e.g., TlCuCl3.
