Over the past few years, there has been a resurgence of interest in using recurrent neural network-hidden Markov model (RNN-HMM) for automatic speech recognition (ASR). Some modern recurrent network models, such as long shortterm memory (LSTM) and simple recurrent unit (SRU), have demonstrated promising results on this task. Recently, several scientific perspectives in the fields of neuroethology and speech production suggest that human speech signals may be represented in discrete point patterns involving acoustic events in the speech signal. Based on this hypothesis, it may pose some challenges for RNN-HMM acoustic modeling: firstly, it arbitrarily discretizes the continuous input into the interval features at a fixed frame rate, which may introduce discretization errors; secondly, the occurrences of such acoustic events are unknown. Furthermore, the training targets of RNN-HMM are obtained from other (inferior) models, giving rise to misalignments. In this paper, we propose a recurrent Poisson process (RPP) which can be seen as a collection of Poisson processes at a series of time intervals in which the intensity evolves according to the RNN hidden states that encode the history of the acoustic signal. It aims at allocating the latent acoustic events in continuous time. Such events are efficiently drawn from the RPP using a sampling-free solution in an analytic form. The speech signal containing latent acoustic events is reconstructed/sampled dynamically from the discretized acoustic features using linear interpolation, in which the weight parameters are estimated from the onset of these events. The above processes are further integrated into an SRU, forming our final model, called recurrent Poisson process unit (RPPU). Experimental evaluations on ASR tasks including ChiME-2, WSJ0 and WSJ0&1 demonstrate the effectiveness and benefits of the RPPU. For example, it achieves a relative WER reduction of 10.7% over state-of-the-art models on WSJ0.