The chromium spinels MgCr_{2}O_{4} and ZnCr_{2}O_{4} are prime examples of the highly frustrated pyrochlore lattice antiferromagnet. Experiment has carefully established that both materials, upon cooling, distort to lower symmetry and order magnetically. We study the nature of this process by a combination of density-functional-theory-based energy mapping and classical Monte Carlo simulations. We first computationally establish precise Heisenberg Hamiltonian parameters for the high temperature cubic and the low temperature tetragonal and orthorhombic structures of both spinels. We then investigate the respective ordering temperatures of high symmetry and low symmetry structures. We carefully compare our results with experimental facts and find that our simulations are remarkably consistent with a type of spin-Peierls mechanism, adapted to three dimensions, where the structural distortion is mediated by a magnetic energy gain due to a lower degree of frustration.