The outstanding accuracy achieved by modern Automatic Speech Recognition (ASR) systems is enabling them to quickly become a mainstream technology. ASR is essential for many applications, such as speech-based assistants, dictation systems and real-time language translation. However, highly accurate ASR systems are computationally expensive, requiring on the order of billions of arithmetic operations to decode each second of audio, which conflicts with a growing interest in deploying ASR on edge devices. On these devices, hardware acceleration is key for achieving acceptable performance. However, ASR is a rich and fast-changing field, and thus, any overly specialized hardware accelerator may quickly become obsolete. In this paper, we tackle those challenges by proposing ASRPU, a programmable accelerator for on-edge ASR. ASRPU contains a pool of general-purpose cores that execute small pieces of parallel code. Each of these programs computes one part of the overall decoder (e.g. a layer in a neural network). The accelerator automates some carefully chosen parts of the decoder to simplify the programming without sacrificing generality. We provide an analysis of a modern ASR system implemented on ASRPU and show that this architecture can achieve real-time decoding with a very low power budget.