Toric geometry has been an indispensable tool in constructing Calabi-Yau manifolds that arise in string compactifications without background fluxes. In this talk, I will show that toric varieties can also be equipped with $G$-structures, and are thereby also candidates for internal manifolds of flux compactifications. After a brief introduction to toric geometry, I will review the (partial) classification of three-dimensional smooth complex toric varieties (SCTV) by Oda. I will then propose a method for constructing $SU(3)$ structure on SCTVs. For some examples, I will discuss the torsion classes of the $SU(3)$ structure, and check if the conditions for supersymmetric string theory vacua are fulfilled.