The Brillouin light scattering (BLS) technique is used to observe thermally excited acoustic surface phonons in backscattering geometry. A series of DC sputtered ITO films with thicknesses ranging from 31 to 3600 nm were deposited at a substrate temperature of 150 degreesC on hydrogen terminated (100) silicon. From the dispersion of the Rayleigh mode and of the higher order Rayleigh-like modes in the discrete part of the spectrum, three independent stiffness constants, c(11), c(33) and c(55), are determined with reasonable accuracy. These results are obtained using a hexagonal model, which takes the axial symmetry of the film material into account. The elastic anisotropy, c(11)/c(33) approximate to 5/4, is attributed to the single crystal anisotropy of the strongly textured ITO. The Young's moduli obtained from BLS and a microindentation analysis are in good agreement.