We present herein 3D elastic tensor imaging (3D ETI), an ultrasound-based volumetric imaging technique to provide quantitative volumetric mapping of tissue elastic properties in weakly elastic anistropic media. The technique relies on (1) 4D ultrafast shear wave elastography (SWE) at very high volume rate (e.g. > 8000 Hz, depending only on the imaging depth), (2) a volumetric estimation of shear wave velocity using the eikonal equation and (3) a generalized 3D elastic tensor-based approach. 3D ETI was first evaluated using numerical simulations in homogeneous isotropic and transverse isotropic media. Results showed that 3D ETI can accurately assess tissue stiffness and tissue anisotropy in weakly transversely isotropic media (elastic fractional anisotropy coefficient < 0.34). Experimental feasibility was shown in vitro in a transverse isotropic phantom. Quantification of the elastic properties by 3D ETI was in good agreement with 2D SWE results performed at different orientations using a clinical ultrafast ultrasound scanner. 3D ETI has the potential to provide a volumetric quantitative map of tissue elastic properties in weakly transversely isotropic soft tissues within less than 20 ms of acquisition for the entire imaged volume.