This work is motivated by the need for software computing 3D periodic triangulations in numerous domains including astronomy, material engineering, biomedical computing, fluid dynamics etc. We design an algorithmic test to check whether a partition of the 3D flat torus actually forms a triangulation (which subsumes that it is a simplicial complex). We propose an incremental algorithm that computes the Delaunay triangulation of a set of points in the 3D flat torus without duplicating any point, whenever possible. As far as we know, this is the first algorithm of this kind whose output is provably correct. Proved algorithms found in the literature are in fact always computing with 27 copies of the input points, while our algorithmic test detects when such a duplication can be avoided, which is usually possible in practical situations. The algorithm was implemented and is going to be submitted to the CGAL Editorial Board. A video of this work was presented at SoCG'08.