Significance The thermal conductivity of suspended graphene can be even higher than the basal-plane value of graphite, which is among the highest found in solids. However, when graphene is in contact with an amorphous material, the thermal conductivity is suppressed considerably. This paper reports that the thickness of multilayer graphene supported on an amorphous substrate needs to be more than 30 atomic layers to recover the graphite thermal conductivity. The finding is explained by long phonon mean free paths in graphite even along the cross-plane direction, and is used to clarify interface leakage of phonons as an important mechanism for the observed suppression. The result is relevant for the application of graphene for electronics, thermal management, and other applications.