We have analytically investigated three-dimensional (3D) Airy-Cartesian (AiC) and Airy-helical-Cartesian (AiHC) wave packets by solving the (3+1)-dimensional free space Schrödinger equation. The complex amplitude of the AiC and AiHC wave packets is described by the product of the Hermite polynomials, the Gaussian functions, and the Airy functions. We find that the AiC and AiHC wave packets will gradually spread along the propagation distances, but attenuate along the time domain. The distributions of the radiation force depend on the distribution factor. In addition, the chirp factor has the effect of the energy concentration of the wave packets. The different 3D spatiotemporal profiles, the photoelasticity, and vortex rings can be obtained by choosing different values of the mode number, the distribution factor, the chirp factor, the initial velocity, and the propagation distances.