As high-power electronic devices are packed to progressively higher densities, synthetic diamond films are being considered as heat spreaders for the prevention of thermal damage (see ref. 1 for example). Although diamond single crystals are known to have the highest thermal conductivity for any material at room temperature (22 W cm-1 K-1 for diamond with natural isotopic abundance, compared with 4 W cm-1 K-1 for copper), the dependence of conductivity on the microstructure of polycrystalline diamond films is not understood. Using a newly developed laser technique2, we have measured thermal conductivity in the experimentally difficult direction perpendicular to the plane of the diamond film. Taken together with earlier in-plane measurements3, this gives a complete description of the local thermal conductivity, showing a significant gradient and anisotropy correlated with the inhomogeneous grain structure. Despite phonon scattering at lattice defects and grain boundaries, we find that the local conductivity near the top growth surface of a synthetic diamond film is, surprisingly, at least as high as that of gem-quality diamond single crystals.