In 2005, Hafting et al. reported that some neurons in the entorhinal cortex (EC) fire bursts when the animal occupies locations oraganized in a hexagonal grid pattern in their spatial environment. Previous to that, place cells had been observed, firing bursts only when the animal occupied a particular region of the environment. Both of these types of cells exhibit theta-cycle modulation, firing bursts in the 4-12Hz range. In particular, grid cells fire bursts of action potentials that precess with respect to the theta cycle, a phenomenon dubbed "theta precession". Since then, various models have been proposed to explain the relationship between grid cells, place cells, and theta precession. However, most models have lacked a fundamental, overarching framework. As a reformulation of the pioneering work of Welday et al. (2011), we propose that the EC is implementing its spatial coding using the Fourier Transform. We show how the Fourier Shift Theorem relates to the phases of velocity-controlled oscillators (VCOs), and propose a model for how various other spatial maps might be implemented (eg. border cells). Our model exhibits the standard EC behaviours: grid cells, place cells, and phase precession, as bourne out by theoretical computations and spiking-neuron simulations. We hope that framing this constellation of phenomena in Fourier Theory will accelerate our understanding of how the EC – and perhaps the hippocampus – encodes spatial information.