  We present an analysis of acoustic phonon propagation through long,
free-standing, insulating wires with rough surfaces. Due to a crossover from
ballistic propagation of the lowest-frequency phonon mode at $\omega <\omega
_{1}=\pi c/W$ to a diffusive (or even localized) behavior upon the increase of
phonon frequency, followed by re-entrance into the quasi-ballistic regime, the
heat conductance of a wire acquires an intermediate tendency to saturate within
the temperature range $T\sim \hbar \omega_{1}/k_{B}$.
