  A class of cross-shaped difference operators on a two dimensional lattice is
introduced. The main feature of the operators in this class is that their
formal eigenvectors consist of multiple orthogonal polynomials. In other words,
this scheme generalizes the classical connection between Jacobi matrices and
orthogonal polynomials to the case of operators on lattices. Furthermore we
also show how to obtain 2D discrete Schr\"odinger operators out of this
construction and give a number of explicit examples based on known families of
multiple orthogonal polynomials.
