We study waveguiding of the electromagnetic energy below the diffraction limit with arrays of dielectric nanoparticles through the excitation of both electric and magnetic Mie resonances. We analyze the dispersion characteristics of such coupled-resonator optical waveguides by means of the coupled-dipole approximation and then verify the validity of the coupled-dipole model by comparing the results with direct numerical simulations. We reveal that a chain of silicon nanoparticles with realistic material losses can guide light for the distances exceeding several tens of micrometers, which is significantly better than the guiding by any plasmonic waveguide with the propagation distances less than 1 μm. We verify the main concept and our theoretical findings experimentally at microwaves for an array of ceramic particles.