The diffraction of Gaussian and arbitrary-order Bessel wavefronts on 2D periodic amplitude and phase gratings, including those with holes whose diameters are close to the wavelength, was investigated. When an amplitude grating is illuminated by a Bessel beam, the self-images of the grating in the main and fractional Talbot planes turn into lattices of rings, whose radii depend in a complex way on the grating parameters and on the illuminating beam characteristics. The results of the experiments carried out using the wavelength-tunable radiation of the Novosibirsk free electron laser (NovoFEL) are in good agreement with the numerical calculations and the analytical model developed.