  We present analytic results for a special dimer model on the {\em
non-bipartite} and {\em non-planar} checkerboard lattice that does not allow
for parallel dimers surrounding diagonal links. We {\em exactly} calculate the
number of closed packed dimer coverings on finite checkerboard lattices under
periodic boundary conditions, and determine all dimer-dimer correlations. The
latter are found to vanish beyond a certain distance. We find that this
solvable model, despite being non-planar, is in close kinship with well-known
paradigm-setting planar counterparts that allow exact mappings to
$\mathbb{Z}_2$ lattice gauge theory.
