  Fruitful ideas on how to quantize gravity are few and far between. In this
paper, we give a complete description of a recently introduced non-perturbative
gravitational path integral whose continuum limit has already been investigated
extensively in d less than 4, with promising results. It is based on a
simplicial regularization of Lorentzian space-times and, most importantly,
possesses a well-defined, non-perturbative Wick rotation. We present a detailed
analysis of the geometric and mathematical properties of the discretized model
in d=3,4. This includes a derivation of Lorentzian simplicial manifold
constraints, the gravitational actions and their Wick rotation. We define a
transfer matrix for the system and show that it leads to a well-defined
self-adjoint Hamiltonian. In view of numerical simulations, we also suggest
sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological
phases found previously in Euclidean models of dynamical triangulations cannot
be realized in the Lorentzian case.
