Ultrasound shear wave elastography is emerging as an important imaging modality for evaluating tissue material properties. In its practice, some systematic biases have been associated with ultrasound frequencies, focal depths and configuration, and transducer types (linear versus curvilinear), along with displacement estimation and shear wave speed estimation algorithms. Added to that, soft tissues are not purely elastic, so shear waves will travel at different speeds depending on their spectral content, which can be modulated by the acoustic radiation force (ARF) excitation focusing, duration, and the frequency-dependent stiffness of the tissue. To understand how these different acquisition and material property parameters may affect the measurements of shear wave velocity, the simulations of the propagation of shear waves generated by ARF excitations in viscoelastic media are a very important tool. This paper serves to provide an in-depth description of how these simulations are performed. The general scheme is broken into three components: 1) simulation of the 3-D ARF push beam; 2) applying that force distribution to a finite-element model; and 3) extraction of the motion data for post-processing. All three components will be described in detail and combined to create a simulation platform that is powerful for developing and testing algorithms for academic and industrial researchers involved in making quantitative shear-wave-based measurements of tissue material properties.