Based on the diffusional scattering of pseudophotons, we propose a new mechanism of x-ray radiation from a relativistic charged particle moving in a system consisting of microspheres distributed randomly in a dielectric material. The mechanism leads to a stronger dependence of radiation intensity on the particle energy, gamma = E/mc(2), than that predicted by the transition radiation, and explains recent experimental data from a detector which contains randomly distributed superconducting granules.