The behaviour of the Ricci flow on hyperbolic surfaces is investigated via a combinatorial analogue first studied by Chow and Luo. The natural cell decompositions of Teichmueller and moduli spaces of hyperbolic metrics on decorated surfaces give parameters for and are shown to be compatible with the combinatorial flow. Hence elegant new proofs of the cell decompositions are obtained, as well as a practical algorithm for reconstructing a metric on a hyperbolic surface from a point in the corresponding cell complex.