We propose a new variant of the celebrated Total Variation image denoising model of Rudin, Osher and Fatemi, which provides results very similar to the Bayesian posterior mean variant (TV-LSE) while showing a much better computational efficiency. This variant is based on an iterative procedure which is proved to converge linearly to a fixed point satisfying a marginal conditional mean property. The implementation is simple, provided numerical precision issues are correctly handled. Experiments show that the proposed variant yields results that are very close to those obtained with TV-LSE and avoids as well the so-called staircasing artifact observed with classical Total Variation denoising.