I think the answer is that every such graph should be quasi-isometric to the infinite line. Look at <a href="http://arxiv.org/PS_cache/math/pdf/9806/9806129v1.pdf"> this paper.</a> Theorem 9 there states that even a quasi-transivite amenable graph should have trivial Poisson boundary (no non-constant harmonic functions). On the other hand, a hyperbolic transitive graph, if it is not quasi-isometric to the Cayley graph $\mathbb Z$, always has non-trivial Poisson boundary. <a href="http://www.cmat.edu.uy/~lessa/tesis/Kaimanovich%20-%20groups%20with%20hyperbolic.pdf">Here</a> it is proved for groups, but I think it works for transitive hyperbolic graphs as well.