||It is well known that the complexity, i.e., the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n<sup>2</sup>). Interest has recently arisen as to what happens, both in deterministic and probabilistic situations, when the three dimensional points are restricted to lie on the surface of a 2-dimensional object. In this paper we consider the situation when the points are drawn from a 2-dimensional Poisson distribution with rate n over a fixed union of triangles in R<sup>3</sup>. We show that with high probability the complexity of their Voronoi diagram is Õ(n). This implies, for example, that the complexity of the Voronoi diagram of points chosen from the surface of a general fixed polyhedron in R<sup>3</sup> will also be Õ(n) with high probability.