Kinematic features of three–dimensional mixing by advection of passive particles in time–periodic flows are the primary subject of this study. A classification of periodic points, providing important information about the mixing properties of a flow, is presented, and the dynamics of the Poincaré map in the vicinity of periodic points is analysed for all identified types. Three examples of Stokes flow in a finite cylindrical cavity with discontinuous periodic motion of its end walls are used to illustrate the determination of both periodic lines and isolated periodic points in the flow domain. The stable and unstable manifolds of points on the periodic lines create two surfaces in the flow. A numerical technique based on tracking of a material surface is presented to study the manifold surfaces and their intersections. It is illustrated with numerical examples that flows with periodic lines possess only quasi–two–dimensional mechanisms of chaotic advection.