As the application of deep learning continues to grow, so does the amount of data used to make predictions. While traditionally, big-data deep learning was constrained by computing performance and off-chip memory bandwidth, a new constraint has emerged: privacy. One solution is homomorphic encryption (HE). Applying HE to the client-cloud model allows cloud services to perform inference directly on the client's encrypted data. While HE can meet privacy constraints, it introduces enormous computational challenges and remains impractically slow in current systems. This paper introduces Cheetah, a set of algorithmic and hardware optimizations for HE DNN inference to achieve plaintext DNN inference speeds. Cheetah proposes HE-parameter tuning optimization and operator scheduling optimizations, which together deliver 79x speedup over the state-of-the-art. However, this still falls short of plaintext inference speeds by almost four orders of magnitude. To bridge the remaining performance gap, Cheetah further proposes an accelerator architecture that, when combined with the algorithmic optimizations, approaches plaintext DNN inference speeds. We evaluate several common neural network models (e.g., ResNet50, VGG16, and AlexNet) and show that plaintext-level HE inference for each is feasible with a custom accelerator consuming 30W and 545mm^2.