Most urethanes exhibit high hysteresis...when compressed, they do not quickly return to the previous dimension. This is not a problem given a low frequency cycle, i.e. plenty of time for dimensional recovery. At higher frequencies, it either maintains the compressed shape (which can put freeplay in an oscillating system, see next post),or if forced to follow the vibration, it heats rapidly.
In terms of vibration isolation, hysteresis makes the spring-mass system highly damped. Although that can be a useful property when the goal is mass control while operating at a resonant frequency, it also makes the material a poor isolator at non-resonant frequencies. This is the issue being discussed in previous posts. Given that resonant operation is not a factor in selecting a Lycoming mount, there isn't much reason to accept the compromise.
Because urethanes are engineered compounds, it is possible that the issue can be minimized...but I suspect the nice folks at the vibration control companies have already been there and done that. Cruise the Lord Aerospace catalog for illustration. In addition to 50 pages of vibration tutorial, you'll notice that the mount materials are are natural rubber, neoprene, and a variety of
silicone compounds.
http://www.lord.com/Documents/Product Catalogs/PC6116_AerospaceandDefenseIsolatorCatalog.pdf
Switch to a torsional coupler catalog, and I think you'll find that urethanes are limited to high stiffness couplings, i.e. applications that will not actually deform the urethane disk, diaphragm, or donut very much under load. It's not because they can't make a softer urethane. It's to limit heating due to hysteresis, engine output frequencies being high enough make it a problem.
Summary? Urethane motor mounts are probably a case of reinventing the square wheel.