Consider a polyhedral surface consisting of n triangular faces where each face has an associated positive weight. The cost of travel through each face is the Euclidean distance traveled multiplied by the weight of the face. We present an approximation algorithm for computing a path such that the ratio of the cost of the computed path with respect to the cost of a shortest path is bounded by (1 + ε), for a given 0 < ε < 1. The algorithm is based on a novel way of discretizing the polyhedral surface. We employ a generic greedy approach for solving shortest path problems in geometric graphs produced by such discretization. We improve upon existing approximation algorithms for computing shortest paths on polyhedral surfaces [1,4,5,10,12,15].