What's wrong with casual dynamical triangulation? Thinking about how quantum field theory is calculated. Specifically, anything any more complicated than $U(1)$ Yang-Mills, one needs to use lattice QCD. The lattice spacing going to zero in the limit.
Now once, you have divided space into a lattice, it doesn't seem that great a step to incorporate curvature by allowing the lattice to become a triangular mesh, and the mesh being a triangulation of curved-space time and then summing over the different meshes. Of course that's still treating gravity a bit too much like a smooth field rather than like interacting gravitons. And I suppose it's not that clear how the particles on the mesh interact with the mesh to make interactions between gravity and matter. (In my mind, a graviton would be like the pentagons you could add to a sheet of hexagons to make it curve.)
This is more or less the approach of 'causal dynamical triangulation'. It seems like the logical way to quantise space-time. But, apparently, it isn't otherwise more people would be working on it.
So I wondered what is fundamentally wrong with this approach? It seems like whichever theory you end up with you would have to do some sort of lattice calculations with it. And has it been solved how matter would interact with a dynamical mesh like this, or is this really treating gravity semi-classically? With a lattice on a classical gravity background?
 A: I'm a PhD student, working on CDT, so maybe I can answer from inside! :)
Firstly, there is a false statement in your question. CDT is not a triangular coarse graining over an existing spacetime, but spacetime is constructed by gluing simplices. This way you can create a manifold, and use Regge calculus to define observables such as curvature, volume, distances and so on.
Big difference with LQCD is background independence. The Lattice in qcd is IN the spacetime, while in CDT it is THE spacetime.
The averaging over spacetime Triangulations comes into picture when we apply Feynman path integral formulation, thus making it a quantum theory of geometry.
Is it simply geometry or also physics? The results of the recent years suggests, that even though almost nothing is fixed by hands in the theory, there is an emergent de-Sitter spacetime with proper background volume fluctuations in agreement with the Hartle-Hawking minisuperspace model.
The most recent results suggests that a CDT universe is a few planck length sized Universe, probably before the inflation, where the seeds of the initial density webs (filaments, voids formed by "dark matter" ) can be observed. Again I state that nothing is put in by hand into the model, its simply the Regge version of Einstein gravity, with the cosmological constant and the scalar curvature in the action.
CDT is one of the most promising theories of quantum gravity. Many of its numerical results are in agreement with other models (asymptotic safety, LQG).
It is solved how matter is added. We have simulations with massless and massive scalar fields and massive particles too. Vector fields are not yet solved though, but it is coming soon.
