We consider an arbitrary 2D periodic structure in the low-frequency li
mit and demonstrate that it exhibits birefringence for propagation in
the plane of periodicity. For polarization parallel to the cylinder ax
is the effective dielectric constant, epsilon(eff) is just the average
of the constituents, as known. On the other hand, for in-plane polari
zation, epsilon(eff) is a function of the propagation direction. Numer
ical calculations have been performed for triangular cylinders forming
a square lattice. This leads to an anisotropy as large as 20%, much i
n excess of anisotropy observed in natural crystals.