  We use a Langevin approach to treat the finite temperature dynamics of
displacement variables in the half-filled spinless Holstein model. Working in
the adiabatic regime we exploit the smallness of the adiabatic parameter to
simplify the memory effects and estimate displacement costs from an
"instantaneous" electronic Hamiltonian. We use a phenomenological damping rate,
and uncorrelated thermal noise. The low temperature state has checkerboard
charge order (CO) and the Langevin scheme generates equilibrium thermodynamic
properties that accurately match Monte Carlo results. It additionally yields
the dynamical structure factor, $D({\bf q}, \omega)$, from the displacement
field $x({\bf r}, t)$. We observe four regimes with increasing temperature,
$T$, classified in relation to the charge ordering temperature, $T_c$, and the
`polaron formation' temperature $T_P$, with $ T_c \ll T_P$. For $T \ll T_c$ the
oscillations are harmonic, leading to dispersive phonons, with increasing $T$
bringing in anharmonic, momentum dependent, corrections. For $T \sim T_c$,
thermal tunneling events of the $x({\bf r})$ field occur, with a propagating
`domain' pattern at wavevector ${\bf q} \sim (\pi, \pi)$ and low energy weight
in $D({\bf q}, \omega)$. When $T_c < T < T_P$, the disordered polaron regime,
domain structures vanish, the dispersion narrows, and low energy weight is
lost. For $T \gtrsim T_P$ we essentially have uncorrelated local oscillations.
We propose simple models to analyse this rich dynamics.
