A new laboratory apparatus measures the complex Young's modulus of rock samples at frequencies between 4 and 400 Hz and at strain amplitudes near 10-7. There is negligible attenuation and modulus dispersion in vacuum-dry rocks. In water-saturated samples of sandstone, limestone, and granite there are large attenuation peaks which are caused by stress relaxations. The attenuation and modulus data can be described by Cole-Cole distributions of relaxation times. The rocks anelasticity therefore satisfied the Kramers--Kronig integral relations. The stress relaxations occur at low frequencies, have narrow distributions of characteristic times, are thermally activated, and have activation energies (16--22 kJ/mol) which are characteristic of hydrogen bonds. The modulus defect and peak attenuation are shown to vary with different pore fluids (water, ethanol, and n-decane). The results are explained in terms of the pore fluids lowering the surface free energy of the rock-forming minerals. The relaxation process involves the movement of fluid molecules. The molecules, by bonding to the surface, reduce the surface energy and thus generate the frequency-dependent softening in the rock.