We discuss a differential geometry approach to modeling the dissipative dynamics of relativistic viscous fluids. In such an approach, the matter (continuous medium) is treated as a non-Riemannian manifold with the anholonomic basis tetrad being the main field of the theory. The tetrad field describes the deformation and rotation of the material elements as well as the associated inertial effects. The resulting system of governing PDEs is causal and represented by the system of first-order hyperbolic equations with relaxation source terms. A few numerical examples on flat and curved spacetimes will be presented in order to illustrate the performance of the new model.