  The N\'eel temperature, $T_{\rm N}$, of quasi-one- and quasi-two-dimensional
antiferromagnetic Heisenberg models on a cubic lattice is calculated by Monte
Carlo simulations as a function of inter-chain (inter-layer) to intra-chain
(intra-layer) coupling $J'/J$ down to $J'/J\simeq 10^{-3}$. We find that
$T_{\rm N}$ obeys a modified random-phase approximation-like relation for small
$J'/J$ with an effective universal renormalized coordination number,
independent of the size of the spin. Empirical formulae describing $T_{\rm N}$
for a wide range of $J'$ and useful for the analysis of experimental
measurements are presented.
