We discuss models for reaction-diffusion phenomena based on hyperbolic equations. The standard approach uses parabolic systems, which are well suited to explain events such as heat transmission in close-to-equilibrium regimes, but are criticizable for several reasons (e.g. the prediction of infinite speed of propagation, the lack of time-delay and related inertial effects, and the exceptionality of well-posed boundary value problems). In many contexts the hyperbolic corrections are relevant for applications: dynamics of biological tissues, population growth, forest fire models. We adopt a description by means of hyperbolic models - starting from the basic example of telegraph equation - which are more appropriate when the relaxation time to perceive changes of the overall phenomenon is sufficiently large as compared to the diffusivity coefficient, and differences emerge in the transient regimes whose cumulations may influence significantly the final outcome. The emphasis is placed on the numerical computation of the propagation speed of special traveling wave solutions, i.e. propagating fronts. Three basic numerical schemes are presented, two of which can also be applied to general hyperbolic systems (with reduced performance when dealing with discontinuous initial data). We compare their performance for providing effective approximations of the propagation speed. We focus on a special class of 2x2 systems corresponding to second order PDEs in 1D as simplified modeling of reaction-diffusion equations with monostable and bistable reaction terms. Beside the phase-plane algorithm for hyperbolic reaction-diffusion systems with damping, we propose two PDE-based numerical schemes, the so-called scout&spot algorithm - based on tracking the level curve of some intermediate value of the wave profile - and the LeVeque-Lee formula - given by the average value of the discrete transport velocity - by assessing their capability in comparative experiments of genuine predictions.