Is there a combinatorial analogue of Ricci flow?
Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popular variational approach (though the two approaches intersect nontrivially).
MR0127372 (23 #B418) Regge, T. General relativity without coordinates. (Italian summary) Nuovo Cimento (10) 19 1961 558–571.
MR1393382 (97k:52022) Cooper, Daryl(1-UCSB); Rivin, Igor(4-WARW-MI) Combinatorial scalar curvature and rigidity of ball packings. Math. Res. Lett. 3 (1996), no. 1, 51–60.
MR2136536 (2006a:53081) Glickenstein, David A maximum principle for combinatorial Yamabe flow. Topology 44 (2005), no. 4, 809–825. (Reviewer: Igor Rivin), 53C44 (52C15)
MR2136535 (2005k:53108) Glickenstein, David A combinatorial Yamabe flow in three dimensions. Topology 44 (2005), no. 4, 791–808. (Reviewer: Igor Rivin), 53C44 (52C15)
MR2100762 (2005m:53122) Luo, Feng Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6 (2004), no. 5, 765–780. (Reviewer: Igor Rivin), 53C44 (53C21)
MR2015261 (2005a:53106) Chow, Bennett; Luo, Feng Combinatorial Ricci flows on surfaces. J. Differential Geom. 63 (2003), no. 1, 97–129. (Reviewer: Igor Rivin), 53C44
arXiv:1010.4070 [pdf, ps, other] Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric Xianfeng David Gu, Ren Guo, Feng Luo, Wei Zeng
arXiv:1005.4648 [pdf, other] Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow W. Zeng, L.M. Lui, F. Luo, J.S. Liu T.F. Chan, S.T. Yau, X.F. Gu
Not yet mentioned is the interesting definition of Ricci curvature by Yann Ollivier, a definition especially suited to discrete spaces, such as graphs. His definition "can be used to define a notion of 'curvature at a given scale' for metric spaces." For example, he shows how the discrete cube $\{ 0,1 \}^n$ behaves like $\mathbb{S}^n$ in having constant positive curvature, and possessing an analog of the Lévy "concentration of measure" (the mass of $\mathbb{S}^n$ is concentrated about its equator).
His definition is used in the recent (April, 2011) paper by Jürgen Jost and Shiping Liu: "Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs."
Here are two primary sources:
Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), No. 3, 810-864.
Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains, in Probabilistic approach to geometry, 343-381, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010.
Update (7Feb13). Noticed this recent posting to the arXiv:
Warner A. Miller, Jonathan R. McDonald, Paul M. Alsing, David Gu, Shing-Tung Yau, "Simplicial Ricci Flow," arXiv:1302.0804
[math.DG].
The following paper
F. Luo, A combinatorial curvature flow for compact 3-manifolds with boundary,
http://arxiv.org/abs/math/0405295 (now published in electronic research announcements, AMS, Volume 11, Pages 12--20)
provides a combinatorial flow for 3-manifolds with boundary consisting of surfaces with negative Euler characteristic. It deals with the convergence of an initial "piecewise-hyperbolic" metric to an actual hyperbolic metric with geodesic boundary. The analogy with Ricci flow is very very mild, but I hope you may be interested in this reference anyway.