In this paper, an analytical theory for the diffraction of a Bessel beam of arbitrary order J l(κ r) on a 2D amplitude grating is presented. The diffraction pattern behind the grating turned out to be more complicated in comparison with the classical Talbot effect observed under illumination by a plane wave. In particular, the patterns in the main and fractional Talbot planes under certain conditions that are found in the article, are lattices of ring microbeams, the diameters of which depend on the period of the grating, the diameter of the illuminating beam, the number of the Talbot plane, and the topological charge l. For the rings near the optical axis, the latter reproduces l of the illuminating beam. The diffraction patterns observed in experiments on the Novosibirsk free electron laser behind gratings with periods of 1 to 6 mm, illuminated by zero- to second-order Bessel beams at a radiation wavelength λ = 141 µm, are in good agreement with both theoretical predictions and numerical calculations. We emphasize that an analytical theory based on the scalar theory of diffraction perfectly describes resulting diffraction patterns with hole diameters down to 0.25 mm, which are less than 2λ, when, formally speaking, the scalar theory becomes incorrect. Since the Laguerre-Gaussian beams can be represented as a superposition of Bessel beams, results of this paper can be applied to analysis of the Talbot effect with the Laguerre-Gaussian beams. A regular lattice of ring-like vortex microbeams, which can be formed in the Talbot planes, can be applied, for example, to creation of a lattice of optical traps, transmission of orbital angular momentum to elements of micromechanical devices, and to launching surface plasmon polaritons on an array of metal wires by 'the end-fire coupling technique'.