This paper introduces a way to create monohedral tilings of the surface of a regular tetrahedron. We design a special plane diagram (or pattern) that can be printed on a single sheet of paper and folded up to be a regular tetrahedron. In fact, if a diagram is a net of a regular tetrahedron, then we can use 4 copies of it to construct another new net of a regular tetrahedron; that is, a tiling of the surface of a regular tetrahedron. Finally, by similar construction modes, we can obtain different surface tilings of a regular tetrahedron with 16 copies of it, 64 copies of it, and so on. Thus we have new procedures for creating artistic surface tilings of a regular tetrahedron.