The Kapitza conductance between isotopically enriched synthetic diamond and Ti, Al, Au, and Pb has been studied using picosecond optical techniques. For Ti and Al, which have Debye temperatures roughly 1/5 of that of diamond, the measured heat flow is in reasonable agreement with calculations of the phonon heat transport based on a simple lattice dynamical model. However, for Au and Pb, which have Debye temperatures 0.07 and 0.04 times that of diamond, the measured Kapitza conductances are as much as 100 times larger than expected, suggesting the existence of a second conductance mechanism.