We have developed and demonstrated an experimental method, based on the picosecond acoustics technique, to perform nondestructive complete mechanical characterization of nanowires, that is, the determination of the complete elasticity tensor. By means of femtosecond pump-probe spectroscopy, coherent acoustic phonons were generated in an ensemble of nanowires and their dynamics was resolved. Specific phonon modes were identified and the detection mechanism was addressed via wavelength dependent experiments. We calculated the exact phonon dispersion relation of the nanowires by fitting the experimentally observed frequencies, thus allowing the extraction of the complete elasticity tensor. The elasticity tensor and the nanowire diameter were determined for zinc blende GaAs nanowires and were found to be in a good agreement with literature data and independent measurements. Finally, we have applied this technique to characterize wurtzite GaAs nanowires, a metastable phase in bulk, for which no experimental values of elastic constants are currently available. Our results agree well with previous first principle calculations. The proposed approach to the complete and nondestructive mechanical characterization of nanowires will allow the efficient mechanical study of new crystal phases emerging in nanostructures, as well as size-dependent properties of nanostructured materials.