The polarization dependence of the photoionization probability was analyzed in the case when a randomly oriented atom is irradiated by two crossing polarized monochromatic photon beams with the same frequency. It was found that the angular distributions of photoelectrons exhibit the effect of circular dichroism (CD), which consists of the dependence of the photoionization probability on the sign of the circular polarization degree of each beam. We demonstrate that the CD effect exists only for coherent crossing photon beams. It was shown that CD effects are strongly dependent on the phase difference between the electric field vectors of the photon beams and have a quite large magnitude. The possibilities of the experimental observation of CD effects are discussed.