There are several approches to incorporate torsion into a theory of gravity.
According to this article, requiring a consistent coupling to Dirac fields leaves us with the Einstein-Cartan approach or the rules of general relativity.
The latter can be motivated by the principle of general covariance (see section IV of this review article), but does not restrict us to torsion-free connections:
Rather, there's a sort of gauge symmetry (see this article) between curvature and torsion, ie a whole class of connections that can be used to reproduce the equations (if not the geometry) of general relativity, with the torsion-free Levi-Civita connection and the curvature-free Weitzenböck connection as limiting cases.
Einstein-Cartan theory comes with new physics and is not without its own problems (see this opinion piece (PDF)), so teleparallel gravity would be the conservative choice.
As teleparallel gravity is a gauge theory and actually does not depend on the equivalence of inertial and gravitational mass, it is conjectured to be more friendly to quantization.
However, I've yet to see a theory of quantum gravity based on teleparallelism.
Assuming this is not just the case because everyone qualified to work on it is busy playing with yarn (strings and loops, you know ;)), what are the problems that one runs into with a teleparallel theory of quantum gravity?
Are they the same ones (renormalizability) that prevent us from quantizing general relativity?
For bonus points, is there any relation of the Teleparallelism/GR equivalence and Gauge/Gravity duality of string theory besides the superficial?