The thermal conductivities of a borosilicate glass and a polycarbonate have been measured in the temperature range 0.04-60 K. Some samples contained well-defined holes to provide an additional source of phonon scattering. The results at low temperatures are consistent with the predictions of the Debye model using experimentally measured sound velocities. There is a sharp decrease in phonon mean free path with increasing frequency so that, at higher temperatures (the "plateau" region), thermal transport is provided predominantly by phonons having frequencies much less that ω~=4kTℏ. For very small hole diameters, diffraction of thermal phonons occurs. The data are compared with theoretical models of thermal transport in noncrystalline materials.