Piecewise linear (PWL) systems can exhibit quite complex behaviours. In this paper, the complementarity framework is used for computing periodic steady-state trajectories belonging to linear time-invariant systems with PWL, possibly set-valued, feedback relations. The computation of the periodic solutions is formulated in terms of a mixed quadratic complementarity problem. Suitable anchor equations are used as problem constraints in order to determine the unknown period and to fix the phase of the steady-state oscillation. The accuracy of the complementarity problem solution is shown through numerical investigations of stable and unstable oscillations exhibited by practical PWL systems: a neural oscillator, a deadzone feedback system, a stick-slip system, a repressilator and a relay feedback system.