We show that the memoryless routing algorithms Greedy Walk, Compass Walk, and all variants of visibility walk based on orientation predicates are asymptotically optimal in the average case on the Delaunay triangulation. More specifically, we consider the Delaunay triangulation of an unbounded Poisson point process of unit rate and demonstrate that the worst-case path between any two vertices inside a domain of area $n$ has a number of steps that is not asymptotically more than the shortest path which exists between those two vertices with probability converging to one (as long as the vertices are sufficiently far apart.) As a corollary, it follows that the worst-case path has $O(\sqrt{n}\,)$ steps in the limiting case, under the same conditions. Our results have applications in routing in mobile networks and also settle a long-standing conjecture in point location using walking algorithms. Our proofs use techniques from percolation theory and stochastic geometry.