Electron tunneling through ordered molecular barriers is found to depend on the tunneling direction. In particular, the transmission probability of an electron incident in the normal direction on a 10 Angstrom water layer strongly polarized in the tunneling direction is 4-5 orders of magnitude larger when it enters from the positive (hydrogen) face than from the negative (oxygen) face, in contrast to continuum dielectric model prediction. This emphasizes the importance of the discrete multidimensional nature of the barrier structure and suggests possible rectifying properties of such barriers, beyond the linear regime.