The training of modern speech processing systems often requires a large amount of simulated room impulse response (RIR) data to generalize well in real-world environments. However, simulating realistic RIR typically requires accurate physical modeling, and the acceleration of such process typically requires certain computational platforms. In this paper, we propose fast random approximation of room impulse response (FRA-RIR) to efficiently generate realistic RIR data without specific computational devices. FRA-RIR replaces the physical simulation by a series of random approximations, which significantly speeds up the simulation process and enables fully on-the-fly simulation when training neural networks. Experiments show that FRA-RIR is not only significantly faster than other existing ISM-based tools on standard platforms, but also improves the performance of speech denoising systems evaluated on real-world RIRs. The implementation is available online.