The stability analysis for periodic motions of a class of harmonically excited single degree of freedom oscillators with piecewise linear characteristics is presented. The common characteristic of these oscillators is that they possess viscous and constant damping properties, which depend on their velocity direction. The presence of constant damping terms in the equation of motion introduces acceleration discontinuities and makes possible the appearance of finite time intervals within the periodic solution where the oscillator is stuck at the same position. Harmonic and subharmonic motions with an arbitrary number of solution pieces are examined. The analysis takes advantage of the fact that the exact solution form for any solution piece included between two consecutive zero velocity values is known. It is based on the derivation of a matrix relation which determines how an arbitrary but small perturbation at the beginning of a periodic solution propagates to the end of a response period. Then, results obtained by bifurcation analysis of the periodic solutions are also presented. At the end, some of the analytical predictions are confirmed by considering an example mechanical model.