  Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real
six-dimensional manifolds, can admit G-structures rendering them suitable for
internal manifolds in supersymmetric flux compactifications. We develop
techniques which allow us to systematically construct G-structures on SCTV and
read off their torsion classes. We illustrate our methods with explicit
examples, one of which consists of an infinite class of toric CP^1 bundles. We
give a self-contained review of the relevant concepts from toric geometry, in
particular the subject of the classification of SCTV in dimensions less or
equal to 3. Our results open up the possibility for a systematic construction
and study of supersymmetric flux vacua based on SCTV.
