In 2D, every polygon can be triangulated without introducing new vertices. In particular, lattice polygons can be divided into lattice triangles.
In 3D, the Schönhardt polyhedron cannot be triangulated into tetrahedra without adding new vertices. What about the following weaker question?: Can every lattice polyhedron be subdivided into lattice tetrahedra, allowing the introduction of new vertices (as long as they are lattice vertices)?