Laser-speckle-contrast shear wave (LSC-SW) imaging is an optical method for tracking the propagation of a transient shear wave. With high spatial resolution and sensitivity in detecting displacements, this method is suitable for performing mechanical measurements in vitro. Here, we present a LSC-SW tomographic imaging system for visualizing the propagating shear wave wavefront in four dimensions [i.e., three-dimensional (3D) space plus time]. The volumetric elasticity distribution of a sample is constructed by estimating the speeds of the shear waves propagating along multiple paths at different angles. The proposed method enables multidirectional estimations of the shear wave speed. The capabilities of the imaging system are demonstrated by evaluating isotropy (both homogeneous and heterogeneous) and anisotropy in semiturbid phantoms. The proposed system is suitable for the mechanical characterization of a 3D cell culture system, such as monitoring changes in fiber orientation during the remodeling of the extracellular matrix that is known to be strongly associated with the progression and characterization of tumors.