Ultrasound and magnetic resonance elastography techniques are used to assess mechanical properties of soft tissues. Tissue stiffness is related to various pathologies such as fibrosis, loss of compliance and cancer. The basic principle of elastography methods is measuring shear wave velocity in tissue due to intrinsic motion or an external source of vibration, and relating the velocity to tissue elasticity. All tissue are inherently viscoelastic and ignoring viscosity biases the velocity-based estimates of elasticity and ignores a potentially important parameter of tissue health. We present Attenuation Measuring Ultrasound Shearwave Elastography (AMUSE), a technique that independently measures both shear wave velocity and attenuation in tissue and therefore allows model-free measurement of tissue elasticity and viscosity. Theoretical basis for AMUSE is first derived and validated in finite element simulations and polyvinyl alcohol and excised liver phantoms. AMUSE is used to measure shear wave velocity and attenuation in 8 transplanted livers in patients with potential acute rejection, and the results were compared with the biopsy findings in a blind study. The comparison showed excellent agreement and suggests that AMUSE can be used to separate livers with acute rejection from livers with no rejection.