We propose a novel formulation of deep networks that do not use dot-product neurons and rely on a hierarchy of voting tables instead, denoted as convolutional tables (CTs), to enable accelerated CPU-based inference. Convolutional layers are the most time-consuming bottleneck in contemporary deep learning techniques, severely limiting their use in the Internet of Things and CPU-based devices. The proposed CT performs a fern operation at each image location: it encodes the location environment into a binary index and uses the index to retrieve the desired local output from a table. The results of multiple tables are combined to derive the final output. The computational complexity of a CT transformation is independent of the patch (filter) size and grows gracefully with the number of channels, outperforming comparable convolutional layers. It is shown to have a better capacity:compute ratio than dot-product neurons, and that deep CT networks exhibit a universal approximation property similar to neural networks. As the transformation involves computing discrete indices, we derive a soft relaxation and gradient-based approach for training the CT hierarchy. Deep CT networks have been experimentally shown to have accuracy comparable to that of CNNs of similar architectures. In the low-compute regime, they enable an error:speed tradeoff superior to alternative efficient CNN architectures.