A general approach to generating three-dimensional nondiffracting spatiotemporal solutions of the linear Schrdinger equation with an Airy-beam time-dependence is reported. A class of accelerating optical pulses with the structure of Airy-Gauss-Kummer vortex beams is obtained. Our results demonstrate that the optical field contributions to the Airy-Gauss-Kummer accelerating optical wave packets of the cylindrical symmetry can be characterized by the radial and angular mode numbers. 2013 Elsevier Inc.