We study diffraction of Bessel vortex beams with topological charges of ±1 and ±2 and a wavelength of 130 µm on two-dimensional amplitude periodic gratings. Results of simulations and experiments at the Novosibirsk Free Electron Laser facility show that there appear periodic patterns in the planes corresponding to the classical main and fractional Talbot planes, but instead of self-images of the holes, there are observed periodic lattices of annular vortex microbeams with topological charges corresponding to the charge of the beam illuminating the grating. The ring diameters are the same for beams with different topological charges, but they are proportional to the grating period and inversely proportional to the diameter of the beam illuminating the grating.