We study monotone persuasion in the linear case, where posterior distributions over states are summarized by their mean. We develop a novel methodological approach to solve the two leading cases where optimal unrestricted signals can be nonmonotone. First, if the objective is s-shaped and the state is discrete, then optimal monotone signals are upper censorship, whereas optimal unrestricted signals may require randomization. Second, if the objective is m-shaped and the state is continuous, then optimal monotone signals are interval disclosure, whereas optimal unrestricted signals may require nonmonotone pooling. Unlike with unrestricted persuasion, the value of monotone persuasion can decrease with the informativeness of the prior. We illustrate our results with an application to media censorship.