As far as I can tell, it's completely rigorous. And I'm a mathematician by training, so I'm more fussy about these things than most.
It's easier to make discrete models rigorous than models that assume spacetime is a continuum. That's the main reason I like discrete models.
In particular, all the 3+1-dimensional spin foam models of quantum gravity I've worked on - various versions of the Barrett-Crane model - are mathematically rigorous and background-free.
The problem is, we haven't gotten good evidence that these spin foam models "work" - namely, that they reduce to general relativity in the limit of distance scales that are large compared to the Planck length.
See my Marseille talk for a taste of the problems:
Since we don't have any experimental evidence concerning quantum gravity, mathematical rigor is one way to make sure we're not playing tennis with the net down. I will be very happy when we get any
rigorously well-defined background-free quantum theory of gravity that works
in the sense defined above.
More precisely: I will be very happy if we get numerical evidence that it works, and ecstatic
if we can mathematically prove
that it works. But since such a model is likely to be nonperturbative, a mathematical proof of this sort might be very difficult. Nobody has even proved confinement in lattice QCD, even though numerical calculations have convinced everyone it's true.