The flow of thin liquid films on rotating surfaces is directly relevant to the coating of discrete objects. To begin understanding how surface topography influences such flows, we consider a model problem in which a thin liquid film flows over a rotating cylinder patterned with a sinusoidal surface topography. Lubrication theory is applied to develop a partial differential equation that governs the film thickness as a function of time and the angular coordinate. Static situations are considered first in order to determine the parameter regime in which the lubrication approximation is expected to be valid. When gravitational forces are relatively weak, cylinder rotation leads to the formation of droplets connected by very thin films. The number of droplets is equal to the pattern frequency at low and high rotation rates, with the droplets located at the pattern troughs at low rotation rates and the pattern crests at high rotation rates. When gravitational forces become significant, the film thickness never reaches a steady state, in contrast to the case of an unpatterned cylinder. The results of this work clearly establish that the flow of thin liquid films on rotating surfaces can be very sensitive to the presence of surface topography.