This paper presents a study of the dynamics of a particle undergoing a directed random walk in a two-dimensional disordered square lattice. We derive the asymptotical behaviors of the coordinate and of the mean square displacement. All the dynamical exponents are calculated both in the normal and the anomalous regimes. It is shown that, as contrasted to the one-dimensional case, the so-called quenched and annealed diffusion "constants" indeed coincide.