This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen-Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process. Numerical experiments are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves, and important evidence of the validity of those results beyond the formal hypotheses is numerically established.