Advances in fabrication technologies are enabling the production of architected materials with unprecedented properties. Most such materials are characterized by a fixed geometry, but in the design of some materials it is possible to incorporate internal mechanisms capable of reconfiguring their spatial architecture, and in this way to enable tunable functionality. Inspired by the structural diversity and foldability of the prismatic geometries that can be constructed using the snapology origami technique, here we introduce a robust design strategy based on space-filling tessellations of polyhedra to create three-dimensional reconfigurable materials comprising a periodic assembly of rigid plates and elastic hinges. Guided by numerical analysis and physical prototypes, we systematically explore the mobility of the designed structures and identify a wide range of qualitatively different deformations and internal rearrangements. Given that the underlying principles are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems.