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We present results for the phase diagram of an SU($N$) generalization of theHeisenberg antiferromagnet on a bipartite three-dimensional anisotropic cubic(tetragonal) lattice as a function of $N$ and the lattice anisotropy $\gamma$.In the "isotropic" $\gamma=1$ cubic limit, we find a transition from N\'{e}elto valence bond solid (VBS) between N=9 and N=10. We follow the N\'{e}el-VBStransition to the limiting cases of $\gamma \ll 1 $ (weakly coupled layers) and$\gamma \gg 1$ (weakly coupled chains). Throughout the phase diagram we find adirect first-order transition from N\'{e}el at small-$N$ to VBS at large-$N$.In the three-dimensional models studied here, we find no evidence for either anintervening spin-liquid "photon" phase or a continuous transition, even closeto the limit $\gamma \ll 1$ where the isolated layers undergo continuousN\'{e}el-VBS transitions.

The N\'{e}el-VBS transition in three-dimensional SU($N$) antiferromagnets