Predicting protein properties such as solvent accessibility and secondary structure from its primary amino acid sequence is an important task in bioinformatics. Recently, a few deep learning models have surpassed the traditional window based multilayer perceptron. Taking inspiration from the image classification domain we propose a deep convolutional neural network architecture, MUST-CNN, to predict protein properties. This architecture uses a novel multilayer shift-and-stitch (MUST) technique to generate fully dense per-position predictions on protein sequences. Our model is significantly simpler than the state-of-the-art, yet achieves better results. By combining MUST and the efficient convolution operation, we can consider far more parameters while retaining very fast prediction speeds. We beat the state-of-the-art performance on two large protein property prediction datasets.