This paper discusses the mean-square displacement for a random walk on a two-dimensional lattice, whose transitions to nearest-neighbor sites are symmetric in the horizontal and vertical directions and depend on the column currently occupied. Under a uniform density condition for the step probabilities it is shown that the horizontal mean-square displacement after n steps is asymptotically proportional to n, and independent of the particular column configuration. The model generalizes that of Seshadri, Lindenberg, and Shuler and the arguments are essentially probabilistic.