We undertake a computational and experimental study of an advanced class of structured beams, partially coherent radially polarized vortex (PCRPV) beams, on propagation through atmospheric turbulence. A computational propagation model is established to simulate this class of beams, and it is used to calculate the average intensity and on-axis scintillation index of PCRPV beams. On comparison with other classes of structured beams, such as partially coherent vortex beams and partially coherent radially polarized beams, it is found that the PCRPV beams, which structure phase, coherence and polarization simultaneously, show marked improvements in atmospheric propagation. The simulation results agree reasonably well with the experimental results. These beams will be useful in free-space optical communications and remote sensing.