  Crepant resolutions of three-dimensional toric Gorenstein singularities are
derived equivalent to noncommutative algebras arising from consistent dimer
models. By choosing a special stability parameter and hence a distinguished
crepant resolution $Y$, this derived equivalence generalises the Fourier-Mukai
transform relating the $G$-Hilbert scheme and the skew group algebra
$\CC[x,y,z]\ast G$ for a finite abelian subgroup of $\SL(3,\CC)$. We show that
this equivalence sends the vertex simples to pure sheaves, except for the zero
vertex which is mapped to the dualising complex of the compact exceptional
locus. This generalises results of Cautis-Logvinenko and Cautis-Craw-Logvinenko
to the dimer setting, though our approach is different in each case. We also
describe some of these pure sheaves explicitly and compute the support of the
remainder, providing a dimer model analogue of results from Logvinenko.
