The mechanical properties of a solid dosage, defined by its granular micro-structure and geometry, play a key role in its dissolution profile and performance. An ultrasonic method for extracting the viscoelastic material properties and granular structure of drug tablet compacts is introduced and its utility is demonstrated for tablet compacts made of microcrystalline cellulose (MCC), lactose monohydrate, and sodium starch glycolate as well as magnesium stearate as lubricant. The approach is based on the effect of viscoelasticity and internal micro-structures on the frequency-dependent attenuation of an ultrasonic wave propagating in a granular medium. The models for viscoelastic (a two-parameter Zener model) and scattering attenuation (Rayleigh model) mechanisms are employed. The material parameters including viscoelastic and scattering parameters (average Young's modulus, stress and strain relaxation time constants, and the Rayleigh scattering material parameter) and grain size distribution with a known distribution profile are extracted by an optimization algorithm based on the least square method. The results also indicate good agreement between experimentally and computationally determined phase and group velocities in compacted samples. It is found that the effects of both attenuation mechanisms are present and the extracted grain size distribution parameters are in good agreement with the optically determined values.