Yes, thre are some alternative experimental techniques for the quick measurements of dissipative parameters under consideration, namely, the method of relaxation spectroscopy using reverse torsion pendulum, the method of dynamic contact indentation.
The last method we have been used for study of biotissues in paper "Method for in vivo estimation of viscoelastic characteristics of skeletal muscules // Russian Journal of Biomechanics. – 2007. – Vol. 11, № 1. – P. 44–53.
Hi. you can design an appropriate test specimen out of your material, that is capable to perform a free decay test (means: you impose some impulse loading and observe the free decay of the oscillation. take care to excite only one mode...). From the measured free decay response (time series of displacement) you can determine the logarithmic decrement as a measure of energy dissipation. the logarithmic decrement can be related to a loss angle or other damping measures under some assumptions. But you get the material properties only for one (or few other) excited natural frequencies; for a more complete characterisation of material properties (e.g. for polymeric materials) dma-tests (dynamic-mechanical analysis) might be more promising...
Regarding FFT based damping identification see following technical review for some ideas. regards