Prediction of microstructure evolution during material processing is essential to control the material properties. Simulation tools for microstructure evolution prediction based on physical concepts are computationally expensive and time-consuming. Therefore, they are not practical when either there is an urgent need for microstructure morphology during the process, or there is a need to generate big microstructure datasets. Essentially, microstructure evolution prediction is a spatiotemporal sequence prediction problem, where the prediction of material microstructure is difficult due to different process histories and chemistry. We propose a Predictive Recurrent Neural Network (PredRNN) model for the microstructure prediction, which extends the inner-layer transition function of memory states in LSTMs to spatiotemporal memory flow. As a case study, we used a dataset from spinodal decomposition simulation of FeCrCo alloy created by the phase-field method for training and predicting future microstructures by previous observations. The results show that the trained network predicts quantitatively accurate microstructure morphologies while it is several orders of magnitude faster than the phase field method. The trained model aims to generate future material microstructures by learning from the historical microstructures, where the visual dynamics are believed to have modular structures that can be learned with compositional subsystems. A PredRNN (https://github.com/thuml/predrnn-pytorch) model was trained to predict Fe-Cr-Co microstructures evolution during the spinodal decomposition.