Tremendous progress has been made in sequential processing with the recent advances in recurrent neural networks. However, recurrent architectures face the challenge of exploding/vanishing gradients during training, and require significant computational resources to execute back-propagation through time. Moreover, large models are typically needed for executing complex sequential tasks. To address these challenges, we propose a novel neuron model that has cosine activation with a time varying component for sequential processing. The proposed neuron provides an efficient building block for projecting sequential inputs into spectral domain, which helps to retain long-term dependencies with minimal extra model parameters and computation. A new type of recurrent network architecture, named Oscillatory Fourier Neural Network, based on the proposed neuron is presented and applied to various types of sequential tasks. We demonstrate that recurrent neural network with the proposed neuron model is mathematically equivalent to a simplified form of discrete Fourier transform applied onto periodical activation. In particular, the computationally intensive back-propagation through time in training is eliminated, leading to faster training while achieving the state of the art inference accuracy in a diverse group of sequential tasks. For instance, applying the proposed model to sentiment analysis on IMDB review dataset reaches 89.4% test accuracy within 5 epochs, accompanied by over 35x reduction in the model size compared to LSTM. The proposed novel RNN architecture is well poised for intelligent sequential processing in resource constrained hardware.