The thermal conductivity of thick‐film diamond prepared by chemical vapor deposition (CVD) has been measured with heat flowing in a direction perpendicular to the plane of the film. A laser flash technique with fast infrared detection has been devised for measurement of thin samples with high conductivity. The conductivity perpendicular to the plane is observed to be at least 50% greater than with heat flowing parallel to the plane. This anisotropy is attributed to low‐quality grain boundaries in the columnar microstructure. The observed dependence of the thermal conductivity on microstructure has important implications for thermal management of microelectronic devices with CVD diamond.