We introduce a new technique for constructing three-dimensional (3D) models of incompressible Riemann S-type ellipsoids and compressible triaxial configurations that share the same velocity field as that of Riemann S-type ellipsoids. Our incompressible models are exact steady-state configurations; our compressible models represent approximate steady-state equilibrium configurations. Models built from this method can be used to study a variety of relevant astrophysical and geophysical problems.