I think you would pick 4 variables, say t,u,v,w to represent the 4 elements in matrix X. Perform the multiplication and set the result equal to C. This should give you four ordinary equations, one for each element. Solve them for t,u,v,w.
It might be easier to find the inverse of both A and B and to multiply the matrix equation AXB=C with A^{-1} from left and B^{-1} from right to get X. (X=A^{-1}CB^{-1})