We consider the problem of observer design for systems with periodic disturbances in the system outputs. Assuming that the period T of the disturbance is known, we introduce the idea of mixing past and present output data to remove the disturbance, by defining a new output as the difference between the original output at time t and at time t-T .We determine the exact conditions under which the system is observable from the redefined output, and specify how to design an observer using regular pole-placement techniques and linear time-invariant analysis. We validate the design through simulations.