We study the Heisenberg exchange couplings in polynuclear transition-metal clusters with strong spin frustration using a variety of theoretical techniques. We present results for a trinuclear Cr-III molecule, a tetranuclear Fe-III complex, and an octanuclear Fe-III molecular magnet. We explore the physics of the exchange couplings in these systems using standard broken-symmetry (BS) techniques and a more recently developed constrained density functional theory (C-DFT) approach. The calculations show that the expected picture of localized spin moments on the metal centers is appropriate, and in each case C-DFT predicts coupling constant values in good agreement with experiment. Furthermore, we demonstrate that all of the C-DFT spin states for a given cluster can be reasonably described by a single Heisenberg Hamiltonian. These findings are significant in part because standard BS calculations are in conflict with the experiments on a number of key points. For example, BS-DFT predicts a doublet (rather than quartet). ground state for the Cr-III cluster while for the Fe-III complexes BS-DFT predicts some of the exchange couplings to be ferromagnetic whereas the experimentally derived couplings are all antiferromagnetic. Furthermore, for BS-DFT the best-fit exchange parameters can depend significantly on the set of spin configurations chosen, For example, by choosing configurations with M-S closer to M-S(max) the BS-DFT couplings can typically be made somewhat closed to the C-DFT and experimental results. Thus, in these cases, our results consistently support the experimental findings.