Modern neural-network-based speech processing systems are typically required to be robust against reverberation, and the training of such systems thus needs a large amount of reverberant data. During the training of the systems, on-the-fly simulation pipeline is nowadays preferred as it allows the model to train on infinite number of data samples without pre-generating and saving them on harddisk. An RIR simulation method thus needs to not only generate more realistic artificial room impulse response (RIR) filters, but also generate them in a fast way to accelerate the training process. Existing RIR simulation tools have proven effective in a wide range of speech processing tasks and neural network architectures, but their usage in on-the-fly simulation pipeline remains questionable due to their computational complexity or the quality of the generated RIR filters. In this paper, we propose FRAM-RIR, a fast random approximation method of the widely-used image-source method (ISM), to efficiently generate realistic multi-channel RIR filters. FRAM-RIR bypasses the explicit calculation of sound propagation paths in ISM-based algorithms by randomly sampling the location and number of reflections of each virtual sound source based on several heuristic assumptions, while still maintains accurate direction-of-arrival (DOA) information of all sound sources. Visualization of oracle beampatterns and directional features shows that FRAM-RIR can generate more realistic RIR filters than existing widely-used ISM-based tools, and experiment results on multi-channel noisy speech separation and dereverberation tasks with a wide range of neural network architectures show that models trained with FRAM-RIR can also achieve on par or better performance on real RIRs compared to other RIR simulation tools with a significantly accelerated training procedure. A Python implementation of FRAM-RIR is released.