This brief considers the existence of periodic behaviors for discrete-time second-order multiagent systems with input saturation constraints. We first consider the case where the agent dynamics is a double integrator and then establish conditions on the feedback gains of the linear consensus control law for achieving periodic behaviors. This, in turn, shows that the previously established sufficient condition for reaching global consensus has a necessary aspect since these two conditions are exclusive. We further consider all other second-order agent dynamics and show that these multiagent systems under the linear consensus law exhibit periodic solutions provided the feedback gains satisfy certain conditions. Simulation results are used to validate the theoretical findings.