Plant fibers such as linen are remarkably stiff materials in the longitudinal direction of the fiber. As plant cell walls are composites made Of Cellulose nanocrystals, the so-called rnicrofibrils, embedded in a disordered matrix, those nanocrystals should exhibit an even higher elastic modulus G. We have determined the elastic properties of cellulose microfibrils via the sound velocities measured by inelastic X-ray scattering (IXS). The IXS technique is particularly sensitive to crystal properties by discriminating the contribution of disordered material. A strong anisotropy is observed, with a much lower elastic modulus perpendicular to the fiber direction (G(1) = 15 GPa) than parallel to it (G(2) = 220 GPa). The latter modulus is considerably higher than all values previously determined and will have a significant impact on models for the elastic properties of cellulose microfibrils and of composites based oil them.