Chalcogenide alloys are attracting considerable interest as phase-change materials. Their elastic properties are investigated here. A film of stoichiometric GeSb2Te4 is amorphous as deposited; under annealing at increasing temperatures it crystallizes into a cubic phase and then into a hexagonal phase. The elastic properties of the three phases have been investigated nondestructively by Brillouin light scattering. Measurements detect frequency dispersion relations of Rayleigh and Sezawa acoustic modes. All the elastic moduli can be derived: they increase, together with mass density, upon transition from the amorphous to the cubic and to the hexagonal phase. The measured moduli provide a benchmark for computational predictions of film properties.