Regular grid subgraphs of maximal girth

The unit-distance graph on the $n$-dimensional integer lattice ${\mathbb{Z}}^n$ is called the $n$-dimensional grid. We attempt to maximize the girth of a $k$-regular (possibly induced) subgraph of the $n$-dimensional grid, and provide examples and bounds for selected values of $n$ and $k$, along with more general results. A few cases involving alternative lattices are also considered.