We study monotonicity testing of Boolean functions over the hypergrid $[n]^d$ and design a non-adaptive tester with $1$-sided error whose query complexity is $\tilde{O}(d^{5/6})\cdot \poly(\log n,1/\eps)$. Previous to our work, the best known testers had query complexity linear in $d$ but independent of $n$. We improve upon these testers as long as $n = 2^{d^{o(1)}}$.
To obtain our results, we work with what we call "the augmented hypergrid", which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.