Abstract An Ornstein–Zernike approximation for the two-body correlation function embodying thermodynamic consistency is applied to a system of classical Heisenberg spins on a three-dimensional lattice. The consistency condition determined in a previous work is supplemented by introducing a simplified expression for the mean-square spin fluctuations. The thermodynamics and the correlations obtained are then compared with approximants based on extrapolation of series expansions and with Monte Carlo simulations. Many properties of the model, including the critical temperature, are very well reproduced by this simple version of the theory, but it shows substantial quantitative error in the critical region, both above the critical temperature and with respect to its rendering of the spontaneous magnetization curve. A less simple but conceptually more satisfactory version of the SCOZA is then developed, but not solved, in which the effects of transverse correlations on the longitudinal susceptibility is included, yielding a more complete and accurate description of the spin-wave properties of the model.