  Light propagation in all-dielectric rod-type metamaterials is studied
theoretically. The electric and magnetic dipole moments of the rods are derived
analytically in the long-wavelength limit. The effective permittivity and
permeability of a square lattice of rods are calculated by homogenizing the
corresponding array of dipoles. The role of dipole resonances in the optical
properties of the rod array is interpreted. This structure is found to exhibit
a true left-handed behavior, confirming previous experiments [L. Peng
\textit{et al.}, Phys. Rev. Lett. \textbf{98}, 157403 (2007)]. A scaling
analysis shows that this effect holds at optical frequencies and can be
obtained by using rods made, for example, of silicon.
