The sizable linear polarization signals produced by the scattering of anisotropic radiation in the core of the Ca I 4227 Å line constitute an important observable for probing the inhomogeneous and dynamic plasma of the lower solar chromosphere. Here we show the results of a three-dimensional (3D) radiative transfer complete frequency redistribution investigation of the line's scattering polarization in a magnetohydrodynamical 3D model of the solar atmosphere. We take into account not only the Hanle effect produced by the model's magnetic field but also the symmetry breaking caused by the horizontal inhomogeneities and macroscopic velocity gradients. The spatial gradients of the horizontal components of the macroscopic velocities produce very significant forward scattering polarization signals without the need of magnetic fields, while the Hanle effect tends to depolarize them at the locations where the model's magnetic field is stronger than about 5 G. The standard 1.5D approximation is found to be unsuitable for understanding the line's scattering polarization, but we introduce a novel improvement to this approximation that produces results in qualitative agreement with the full 3D results. The instrumental degradation of the calculated polarization signals is also investigated, showing what we can expect to observe with the Visible Spectro-Polarimeter at the upcoming Daniel K. Inouye Solar Telescope.