We present a three-periodic, chiral, tensegrity structure and demonstrate that it is auxetic. Our tensegrity structure is constructed using the chiral symmetry Π+ cylinder packing, transforming cylinders to elastic elements and cylinder contacts to incompressible rods. The resulting structure displays local reentrant geometry at its vertices and is shown to be auxetic when modeled as an equilibrium configuration of spatial constraints subject to a quasi-static deformation. When the structure is subsequently modeled as a lattice material with elastic elements, the auxetic behavior is again confirmed through finite element modeling. The cubic symmetry of the original structure means that the auxetic behavior is observed in both perpendicular directions and is close to isotropic in magnitude. This structure could be the simplest three-dimensional analog to the two-dimensional reentrant honeycomb. This, alongside the chirality of the structure, makes it an interesting design target for multifunctional materials.