In this work, we propose an azimuthally-invariant periodic leaky-wave (LW) radiator for the generation of Bessel beams and X-waves by means of backward cylindrical LWs at millimeter wavelengths. A rigorous framework is first outlined to understand the theoretical constraints of such a novel design. A specific attention is devoted to the impact of the attenuation constant on the focusing properties of the generated Bessel beams. On this basis, a practical design is then proposed to meet the previous requirements. Numerical results for different frequency spectra confirm the interesting capabilities of the considered structure, paving the way for the first generation of nondiffracting pulses produced by focusing LWs.