  Thermal conductivity measurements were performed on single crystal samples of
the superconducting filled skutterudite compounds PrOs$_4$Sb$_{12}$ and
PrRu$_4$Sb$_{12}$ both as a function of temperature and magnetic field applied
perpendicular to the heat current. In zero magnetic field, the low temperature
electronic thermal conductivity of PrRu$_4$Sb$_{12}$ is vanishingly small,
consistent with a fully-gapped Fermi surface. For PrOs$_4$Sb$_{12}$, however,
we find clear evidence for residual electronic conduction as the temperature
tends to zero Kelvin which is consistent with the presence of nodes in the
superconducting energy gap. The field dependence of the electronic conductivity
for both compounds shows a rapid rise immediately above H$_{c1}$ and
significant structure over the entire vortex state. In the fully gapped
superconductor PrRu$_4$Sb$_{12}$, this is interpreted in terms of multi-band
effects. In PrOs$_4$Sb$_{12}$, we consider the Doppler shift of nodal
quasiparticles at low fields and multiband effects at higher fields.
