We discuss a new heuristic for solving the Euclidean Traveling Salesman Problem (ETSP).  This heuristic is a convex hull-based method and makes use of the Delaunay triangulation of the set of cities to compute a tour for the given set of cities. We conjecture that the expected running time for this algorithm is O(N log N), implying that a new and faster ETSP heuristic is now available