The thermal conductivity kappa of natural, gem-quality diamond, which can be as high as 2500 Wm(-1) K-1 at 25 degrees C, is the highest of any known material. Synthetic diamond grown by chemical vapor deposition (CVD) of films up to 1 mm thick exhibits generally lower values of kappa, but under optimal growth conditions it can rival gem-quality diamond with values up to 2200 Wm(-1) K-1. However, it is polycrystalline and exhibits a columnar microstructure. Measurements on Free-standing CVD diamond, with a thickness in the range 25-400 mu m, reveal a strong gradient in thermal conductivity as a function of position z from the substrate surface as well as a pronounced anisotropy with respect to z. The temperature dependence of kappa in the range 4 to 400 K has been analyzed to determine the types and numbers of phonon scattering centers as a function of z. The defect structure, and therefore the thermal conductivity, are both correlated with the microstructure. Because of the high conductivity of diamond, these samples are thermally thin. For example, laser flash data for a 25-mu m-thick diamond sample is expected to be virtually the same as laser flash data for a 1-mu m-thick fused silica sample. Several of the techniques described here for diamond are therefore applicable to much thinner samples of more ordinary material.