In this paper the uniform pointwise convergence of the discrete ordinate method for weak and strong solutions of the time dependent, linear transport equation posed in a multidimensional, rectangular parallelepiped with partially reflecting walls is established. The first result is that a sequence of discrete ordinate solutions converges uniformly on the quadrature points to a solution of the continuous problem provided that the corresponding sequence of truncation errors for the solution of the continuous problem converges to zero in the saw manner. The second result is that continuity of the solution with respect to the velocity variables guarantees that the truncation errors in the quadrature formula go to zero and hence that the discrete ordinate approximations converge to the solution of the continuous problem as the discrete ordinates become dense. An existence theorem for strong solutions of the continuous problem follows as a result.