The currently available distribution-free confidence interval for a difference of medians in a within-subjects design requires an unrealistic assumption of identical distribution shapes. A confidence interval for a general linear function of medians is proposed for within-subjects designs that do not assume identical distribution shapes. The proposed method can be combined with a method for linear functions of independent medians to provide a confidence interval for a linear function of medians in mixed designs. Simulation results show that the proposed methods have good small-sample properties under a wide range of conditions. The proposed methods are illustrated with examples, and R functions that implement the new methods are provided.