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In this paper, we present a novel passive millimeter-wave (PMMW) imaging system designed using compressive sensing principles. We employ randomly encoded masks at the focal plane of the PMMW imager to acquire incoherent measurements of the imaged scene. We develop a Bayesian reconstruction algorithm to estimate the original image from these measurements, where the sparsity inherent to typical PMMW images is efficiently exploited. Comparisons with other existing reconstruction methods show that the proposed reconstruction algorithm provides higher quality image estimates. Finally, we demonstrate with simulations using real PMMW images that the imaging duration can be dramatically reduced by acquiring only a few measurements compared to the size of the image.