We predict self-collimation and superdiffusion of light propagating in materials with periodically modulatedgain/loss profile. The modulation is parallel and transverse to the light propagation direction and its periodicity ison the wavelength scale. The self-collimation (or diffraction-free propagation) of the beams is found, which issimilar to that observed in photonic crystals. The superdiffusion is also predicted, which means a dependence ofthe beam amplification on the propagation direction, which results in the spatial frequency filtering of the beams.We justify the predictions by the numerical simulations of the paraxial wave propagation equations.