Multiple orthogonal polynomials are the polynomials in one variable which satisfy the orthogonality relations with respect to r>1 different weight functions. We consider the weights which are the products of the classical discrete weights on uniform lattices with noninteger shifts. We show that the weights satisfy a difference Pearson equation and the corresponding multiple orthogonal polynomials have explicit Rodrigues formula. For the case of two weights (r=2) we describe nine families of multiple discrete orthogonal polynomials which have quite a few properties similar to the classical orthogonal polynomials of Charlie, Meixner, Krawtchouk and Hanh. We also discuss some open research problems and some applications.