The papers to read to get started:

A discrete history of the Lorentzian path integral

Reconstructing the Universe

Then the usual:

Dynamically Triangulating Lorentzian Quantum Gravity

Emergence of a 4D World from Causal Quantum Gravity

Semiclassical Universe from First Principles

Spectral dimension of the Universe

(Thanks to this post from Marcus)

And finally, the current literature on ArXiv.