We analyze and experimentally demonstrate the possibility of generating X-waves at microwave and millimeter-wave frequencies by means of a partially open radial parallel-plate waveguide antenna. The structure is azimuthally symmetric and fed in the center by means of a simple vertical coaxial probe, which excites a cylindrical leaky wave. Radially periodic annular slots etched in the upper metal plate allow the propagation of a backward leaky wave, as required for generating Bessel beams in the near-field region. Since X-wavesare polychromatic superpositions of Bessel beams, the wavenumber frequency dispersion of the relevant leaky mode is accounted for by the antenna design. In particular, a dispersion-engineering approach is used to properly select the operating fractional bandwidth for the antenna. Even if the beneficial effect of a large bandwidth is partially neutralized by the dispersive character of the resulting pulse, this being more promient as the bandwidth increases, the experimental results demonstrate the capability of this simple planar design of generating X-waves in the microwave regime. The antenna can be of interest for the design of next-generation medical imaging devices, for non-destructive evaluations, as well as for wideband near-field secure communications and wireless power transfer systems at microwaves and millimeter waves.