  We introduce finite ramified self-affine substrates in two dimensions with a
set of appropriate hopping rates between nearest-neighbor sites, where the
diffusion of a single random walk presents an anomalous {\it anisotropic}
behavior modulated by log-periodic oscillations. The anisotropy is revealed by
two different random walk exponents, $\nu_x$ and $\nu_y$, in the {\it x} and
{\it y} direction, respectively. The values of these exponents, as well as the
period of the oscillation, are analytically obtained and confirmed by Monte
Carlo simulations.
