  The elementary excitations of the spin-ice materials Ho$_2$Ti$_2$O$_7$ and
Dy$_2$Ti$_2$O$_7$ in zero field can be described as independent magnetic
monopoles. We investigate the influence of these exotic excitations on the heat
transport by measuring the magnetic-field dependent thermal conductivity
$\kappa $. Additional measurements on the highly dilute reference compounds
HoYTi$_2$O$_7$ and DyYTi$_2$O$_7$ enable us to separate $\kappa $ into a sum of
phononic ($\kappa_{ph}$) and magnetic ($\kappa_{mag}$) contributions. For both
spin-ice materials, we derive significant zero-field contributions
$\kappa_{mag}$, which are rapidly suppressed in finite magnetic fields.
Moreover, $\kappa_{mag}$ sensitively depends on the scattering of phonons by
magnetic excitations, which is rather different for the Ho- and the Dy-based
materials and, as a further consequence, the respective magnetic-field
dependent changes $\kappa_{ph}(B)$ are even of opposite signs.
