Computer simulation of reactive transport in heterogeneous systems remains a challenge due to the multiscale nature of reactive dynamics and the non-Fickian behavior of transport. This study develops a fully Lagrangian approach via particle tracking to describe the reactive transport controlled by the tempered super- or subdiffusion. In the particle-tracking algorithm, the local-scale reaction is affected by the interaction radius between adjacent reactants, whose motion can be simulated by the Langevin equations corresponding to the tempered stable models. Lagrangian simulation results show that the transient superdiffusion enhances the reaction by enhancing the degree of mixing of the reactants. The proposed particle-tracking scheme can also be extended conveniently to multiscale superdiffusion. For the case of transient subdiffusion, the trapping of solutes in the immobile phase can either decrease or accelerate the reaction rate, depending on the initial condition of the reactant particles. Further practical applications show that the new solver efficiently captures bimolecular reactions observed in laboratories.