Mixed strategy Nash equilibria in $3×3$ games

Question

I am confused as to how one would go about finding the Nash equilibria of a game like this. I know that it requires eliminating some of these strategies by showing that they are strictly dominated by mixes of the other two. I am not sure how to go about this elimination though. Is there a way to find the Nash equilibria of this game and others like it that is more systematic than just guessing which pairs of strategies will appear in the Nash equilibrium? That is, how do I find which strategies are strictly dominated by a mixed strategy and how do I show that this is the case? $$\begin{array} \\&L&C&R\\ T&2,2&2,3&1,2\\ M&0,3&3,2&1,1\\ B&3,1&0,2&0,3\end{array}$$