The training of modern speech processing systems often requires a large amount of simulated room impulse response (RIR) data in order to allow the systems to generalize well in real-world, reverberant environments. However, simulating realistic RIR data typically requires accurate physical modeling, and the acceleration of such simulation process typically requires certain computational platforms such as a graphics processing unit (GPU). In this paper, we propose FRA-RIR, a fast random approximation method of the widely-used image-source method (ISM), to efficiently generate realistic RIR data without specific computational devices. FRA-RIR replaces the physical simulation in the standard ISM by a series of random approximations, which significantly speeds up the simulation process and enables its application in on-the-fly data generation pipelines. Experiments show that FRA-RIR can not only be significantly faster than other existing ISM-based RIR simulation tools on standard computational platforms, but also improves the performance of speech denoising systems evaluated on real-world RIR when trained with simulated RIR. A Python implementation of FRA-RIR is available online\footnote{\url{https://github.com/yluo42/FRA-RIR}}.