I'm not 100% sure that this would do the trick, but off the top of my head you could boil a conductor in water log enough for it to reach thermal equilibrium (so that it's temp will be ~100C throughout) ad then quickly transfer it into a container of room temperature water and take measurements of the water's temp as a function of time. The heat lost by the conductor will approximately equal the heat gained by the water and since you know the specific heat of water, you can calculate how much heat is lost as a function of time based on your temperature measurements. Then simply repeat the process with an insulator of similar dimensions and compare your results.