We construct Heisenberg antiferromagnetic models in arbitrary dimensions that have isotropic valence-bond crystals (VBC's) as their exact ground states. The $d=2$ model is the Shastry-Sutherland model. In the three-dimensional case we show that it is possible to have a lattice structure, analogous to that of ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2},$ where the stronger bonds are associated with shorter bond lengths. A dimer mean-field theory becomes exact at $\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}$ and a systematic $1/d$ expansion can be developed about it. We study the N\'eel-VBC transition at large d and find that the transition is first order in even but second order in odd dimensions.