What goes wrong with GR as a lattice gauge theory? If one tried to formulate General Realativity in a similar manner to say lattice QCD, what goes wrong that makes it not work?
My first thoughts are that for any particular grid, we might have a metric interpreted as the length of a link in the grid. But in normal lattice QCD we assume that the distances between points in the grid are all the same length.
So it would seem that in a lattice description of GR the coarseness of the grid would change from place to place. Thus the "continuum limit" would be approached at different speeds at different areas of the grid. 
I think in lattice QCD you take grids at different sizes and then try to extrapolate the continuum limit. But if the coarseness of the grid changes from place to place due to a metric this extrapolation can't take place. 
Is this an explanation of why trying a lattice formulation of GR would fail? And are there ways round this? i.e. is there a description of lattice gauge theory where the coarseness of the grid need not be uniform?
 A: The problem is deeper and has to do with renormalizability. In any lattice action you have a number of tuneable parameters that influence the long-distance physics. In the plaquette action of lattice QCD we have for instance the dimensionless coupling $\beta$, which is basically a proxy for the ratio between the pion mass and the lattice spacing. When we take the continuum limit (lattice spacing $a \to 0$), we adjust $\beta$ such that the long-distance physics stays the same. The fact that this is possible at all is a non-trivial fact! Roughly speaking we have
$$
\beta \sim 1/g_\text{YM}^2
$$
so $\beta$ blows up as $a \to 0$.
The case of gravity is qualitatively very different. GR is not renormalizable. In practice, it means that you cannot just adjust one parameter (like $\beta$) whilst keeping the long-distance physics (e.g. Newton's law) invariant. If you tried to make $a$ small, you'd realize that you'd have to add more and more terms to achieve this. At best you can restrict yourself to simulating an "effective" version of GR, where you try to make predictions about intermediate scales that somewhere between one parsec and the Planck scale. There are field-theoretical versions of studying effective theories, see e.g. HQET which is intimately related to lattice QCD.
