  We report a study of the thermal conductivity $\kappa$ of the spin-ice
material Dy$_2$Ti$_2$O$_7$. From the anisotropic magnetic-field dependence of
\kappa$ and by additional measurements on the phononic reference compounds
Y$_2$Ti$_2$O$_7$ and DyYTi$_2$O$_7$, we are able to separate the phononic and
the magnetic contributions to the total heat transport, i.e. $\kappa_{ph}$ and
$\kappa_{mag}$, respectively, which both depend on the magnetic field. The
field dependent $\kappa_{ph}$ arises from lattice distortions due to
magnetic-field induced torques on the non-collinear magnetic moments of the Dy
ions. For $\kappa_{mag}$, we observe a highly anisotropic magnetic-field
dependence, which correlates with the corresponding magnetization data
reflecting the different magnetic-field induced spin-ice ground states. The
magnitude of $\kappa_{mag}$ increases with the degree of the ground-state
degeneracy. This anisotropic field dependence as well as various hysteresis
effects suggest that $\kappa_{mag}$ is essentially determined by the mobility
of the magnetic monopole excitations in spin ice.
