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