  In this paper, we establish the existence and stability properties of odd
periodic waves related to the Klein-Gordon type equations, which include the
well known $\phi^4$ and $\phi^6$ models. Existence of periodic waves is
determined by using a general planar theory of ODE. The spectral analysis for
the corresponding linearized operator is established using the monotonicity of
the period map combined with an improvement of the standard Floquet theory.
Orbital stability in the odd sector of the energy space is proved using
exclusively the monotonicity of the period map. The orbital instability of
explicit solutions for the $\phi^4$ and $\phi^6$ models is presented using the
abstract approach in \cite{grillakis1}.
