This paper presents a new approach to sampling images in which samples are taken on a cutset with respect to a graphical image model. The cutsets considered are Manhattan grids, for example every N th row and column of the image. Cutset sampling is motivated mainly by applications with physical constraints, e.g. a ship taking water samples along its path, but also by the fact that dense sampling along lines might permit better reconstruction of edges than conventional sampling at the same density. The main challenge in cutset sampling lies in the reconstruction of the unsampled blocks. As a first investigation, this paper uses segmentation followed by linear estimation. First, the ACA method [1] is modified to segment the cutset, followed by a binary Markov random field (MRF) inspired segmentation of the unsampled blocks. Finally, block interiors are estimated from the pixels on their boundaries, as well as their segmentation, with methods that include a generalization of bilinear interpolation and linear MMSE methods based on Gaussian MRF models or separable autocorrelation models. The resulting reconstructions are comparable to those obtained with conventional sampling at higher sampling densities, but not generally as good as conventional sampling at lower rates.