By solving the (3+ 1) D free-space Schrödinger equation in polar coordinates analytically, we have investigated the propagation of 3D controllably accelerating and decelerating Airy–Bessel–Gaussian (CAiBG) wave packets, even CAiBG wave packets, odd CAiBG wave packets and the superposition of several CAiBG wave packets in free space. The CAiBG wave packets are constructed with the Airy pulses with initial velocity in temporal domain and the Bessel–Gaussian beams in space domain. Due to the initial velocity on Airy pulses, we can obtain decelerating and accelerating Airy–Bessel–Gaussian wave packets by selecting different initial velocities. Moreover, by superposing several CAiBG wave packets, we can obtain the rotating wave packets.