This can be done if you extend the triangulation of your 3-manifold $M$ to a triangulation of a 4-manifold $W$ whose boundary is $M$. You can find the basic idea in Chapter 9 of these draft notes on my web page, and there are very explicit details in this arXiv preprint (with Z. Wang). (In the latter paper we use cubes instead of simplices to make computer implementation easier.) There's also a summary in the notes from a talk.
The main idea is that the Witten-Reshetikhin-Turaev TQFT can be reinterpreted as a 3+1-dimensional TQFT, and the 3+1-dimensional version is "fully extended" (goes all the way down to points). Once you have a fully extended TQFT, standard techniques allow you to construct a state sum model. In this case, the state sum you get is a modified version of the Crane-Yetter state sum. (You would get a Turaev-Viro state sum for a fully extended 2+1 dimensional TQFT.)