In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same result that did not rely on the Riemann mapping theorem.
Start with a simply connected domain, fill it with coins making an hexagonal pattern (or any other packing). Add an extra vertex to their intersection graph, that will correspond to the boundary. Now apply the Koebe-Andreev-Thurston theorem to this planar graph. This defines a map from the centres of the coins to the centres of some circle packing, filling in each triangle by affine maps gives a map that converges to a conformal one.
See https://en.wikipedia.org/wiki/Circle_packing_theorem for details.
Can this approach be generalised to to obtain the full uniformization theorem? https://en.wikipedia.org/wiki/Uniformization_theorem
Is there some discouraging reason why this might be false or hard?