Empirically, it boils down permittivity and conductance of the material. I am not familiar with the physical factors that influence that. For real applications there are other important factors too though. If your substrate gets moldy in a week then it would be hard to package and sell it to a customer.
Certainly. Those parameters and the geometry they're in can be distilled into impedance (see: S-parameters). The larger the difference in impedance, the less power gets through (see: return loss). For microwave ovens you only really care about material properties at 2.4 GHz, since that is where the power is. I'm under the impression this was chosen because water has a resonance at this frequency, and absorbs a lot of power.
> I wonder if there's any organic materials that would work even better.
PCBs are using mostly organic resin. Mechanical constraints limit many otherwise excellent dielectrics.
Also the cheese seems to have quite high loss tangent here. A quick test to this theory at 2.4 GHz involves a microwave oven, thermometer, pieces of RF laminate and dry cheese.