We have grown thin films of hexagonal boron nitride (h-BN) of thickness 1-10 ML on fiber-oriented Ag buffer films on SrTiO₃(001) by pulsed laser deposition. We used Ag buffer films of 40nm thickness to substitute for expensive single crystal metallic substrates. Reflection high-energy electron diffraction (RHEED) was used to monitor the surface structure of the Ag films and to observe the formation of the characteristic h-BN diffraction pattern. Attenuated total reflectance spectroscopy showed the characteristic h-BN peaks at 780 cm^-1 and 1367.4 cm^-1. Ex-situ photoelectron spectroscopy showed that the surface of the h-BN films was stoichiometric. Scanning electron microscopy showed that the h-BN films grew as large, sub-millimeter sheets with nano- and micro-sheets scattered on the surface. The h-BN sheets was easily exfoliated by the micromechanical adhesive tape method. The use of thin film Ag allowed us to adjust the surface morphology of the thin film prior to h-BN growth.