Atoms moving within the spatially periodic potential generated by interfering plane waves of light can mimic the behavior of electrons that move within the spatially periodic potential within a solid-state crystal.  In this manner, neutral atomic gases trapped in optical lattices can serve as an experimental quantum simulator for the physics models that are used to describe electronic materials.  I will describe how one can use optical lattices formed using two different colors of light to produce an array of periodic potentials with differing geometry – specifically, two-dimensional lattices with triangular unit cells – and then how interacts modify the correlations of neutral bosonic atoms trapped within those potentials.  I will summarize two major experimental findings: a quantitative test of the Bose-Hubbard model achieved by comparing gases trapped in two different trapping geometries, and a characterization of inversion-asymmetric spatial correlations in a trimerized (breathing) kagome lattice.  I will conclude with several suggestions for future work.