To successfully use chemical vapour deposition (CVD) diamond thin films in high temperature applications it is important to know their thermal expansion properties. The thermal expansion properties of natural single crystalline diamond have been the subject of many studies. However measuring these properties can be very difficult, especially for CVD diamond films due to the polycrystalline nature of these films. In this article it is proposed to use Raman shift measurements to determine the linear thermal expansion coefficient of diamond. A theoretical relation, based on the Gruneisen basic definition, to correlate high temperature Raman scattering data and linear thermal expansion data, will be derived. The verification of this relation, using literature data on Raman shifts and linear thermal expansion of natural diamond, is presented. The results will be applied to a series of new Raman shift measurements, obtained on flame and hot filament CVD diamond films, in the temperature range 20-600 degrees C.