There have been previous accounts of geometric configurations of linked rings that are periodic in three dimensions. In particular three examples of space-filling toroids derived from regular lattices are known. A fourth is described here. In two of the three known examples the toroids can be replaced by circular tori, which is also possible with the fourth. A completely different kind of triply periodic link can be created using an approach based on Holden’s polylinks, which are finite polyhedral arrangements of linked polygons. The extension to infinite polyhedra has not been considered before.