A new method of X-ray photon correlation spectroscopy (XPCS) is applied for probing the dynamics of surface height fluctuations as a function of lateral length scale in supported polymer films. The short wavelength and slow time scales characteristic of XPCS extend the phase space accessible to scattering studies beyond some restrictions by light and neutron. Measurements were carried out on polystyrene films of thicknesses ranging from 84 to 333 nm at temperatures above the PS glass transition temperature. We present the experimental verification of the theoretical predictions for the thickness, wave vector and temperature dependence of the capillary wave relaxation times for supported polymeric films above the glass transition temperature.