Using the Boltzmann equation techniques, we develop a theory of the planar acoustomagnetoelectric (AME) effect in three-dimensional (3D) gapless Dirac materials with a linear (massless) dispersion law of conduction electrons. The effect arises if the magnetic field H applied to the sample makes an angle Φ≠0, π/2 with the wavevector q of the acoustic wave and consists in the appearance of a dc electric field Eac directed perpendicular to the wavevector q, with all three vectors q, H, and Eac lying in the same plane. We study this effect in the quantum regime (the electron mean free path l0 is large compared to the wavelength 2π/q), where it occurs as a result of the momentum transfer from an excited acoustic wave, considered a flow of individual acoustic quanta, to conduction electrons subjected to the magnetic field. Our theory predicts that for the 3D Dirac material Cd3As2 exposed to a strong, but non-quantizing magnetic field H=10 kOe and an acoustic wave with a frequency of 10 GHz and an intensity of 2 kW/cm2, the AME field Eac with its specific angular dependence (Eac∝sin2Φ) can reach values of the order of 0.01 V/cm at room temperature, which can be readily measured in the experiment.