This is so strange. Just 2 days ago I realised that although excessive sound production is usually a symptom of a machine that is not working properly, if a machine is designed to produce sound to dissipate energy, that energy would not be converted into heat within the machine. The energy is carried away at the speed of sound. I have been thinking about using a similar process to make a "middle of the room" air cond…
Maybe relevant to your nutty idea: Heat is just motion on a small scale. In solid state materials, just like there are quantized configurations of electrons, there are also quantized modes of motion. These are called "phonons". (Analagous to electron / hole model of solid state electronic structure). (It's also true for single molecules, and this is what low energy IR spectroscopy probes. But less relevant for bulk behavior in a liquid or gas state). If the material is anisotropic, probably there will be anisotropic phonons too. The existence / density of thermal modes at IR energies can also determine how much heat a material radiates into the environment versus conducts through its bulk.
SO, just like materials can have an electronic bandgap, materials can have a thermal bandgap too!!! But I can no longer envision what a thermal transistor would behave like. And I'm not sure how this relates to the electronically tuned thermal materials which show you when you google "theal transistor"