We propose an idea for reconstructing 'blocky' conductivity profiles in electrical impedance tomography. By 'blocky' profiles, we mean functions that are piecewise constant, and hence have sharply defined edges. The method is based on selecting a conductivity distribution that has the least total variation from all conductivities that are consistent with the measured data. We provide some motivation for this approach and formulate a computationally feasible problem for the linearized version of the impedance tomography problem. A simple gradient descent-type minimization algorithm, closely related to recent work on noise and blur removal in image processing via non-linear diffusion is described. The potential of the method is demonstrated in several numerical experiments.