Using a local thermometry technique, we have been able to quantitatively measure the thermal resistance R-T of diffusive Andreev interferometers. We find that R-T is strongly enhanced from its normal-state value at low temperatures, and behaves nonlinearly as a function of the thermal current through the sample. We also find that R-T oscillates as a function of magnetic flux with a fundamental period corresponding to one flux quantum Phi(0)=h/2e, demonstrating the phase-coherent nature of thermal transport in these devices. The magnitude of R-T is larger than predicted by recent numerical simulations.