Understanding Heat Heat or thermal energy as understood is some kind of vibrations of molecules / atoms of the matter. If the molecules are tightly bound in case of solids, it is to-and-fro motion what we call vibration, or, it may be random motion of molecules in case of liquids/gases/plasma.
Sound, being waves, is also a vibration of matter. Why, then, if we heat one end of a solid rod, assuming rod is at least few feet in length, does it take ages for the heat to reach the other end, whereas sound reaches in no time ? (sound travels at 1400 m/s approx in solid)
Doesn't it show that heat is more an intra-atomic feature rather than an atomic or molecular motion? Given the fact that electrical good conductors are also good conductors for heat, can we assume that heat is chaotic motion of electrons (the "electron gas") or some other sub-atomic particle? The model should be correlated or validated for all the phenomena that involve heat, some of them listed below:

*

*Solid melts as it is heated, liquid vaporizes when heated.

*Hot material emits light (frequency of which depends upon temperature)

*Light is absorbed converting to heat.

*Microwave produces enormous heating in certain material (eg. a pan of water in a microwave oven)

*Throttling of a gas through a nozzle produces cooling (or absorbs heat)

*Mechanical friction produces heating

*Compression of gas produces heat

*Heating causes expansion in solid, liquid and gas (though mechanism may differ among the 3 states)

*Heat diff can produce an EMF and vice-versa in a thermocouple
(Seebeck / Peltier effect)

*Expansion of rubber band produces heat, contraction absorbs heat

*Passage of electric current through metal produces heat

*Magnetic hysteresis produces heat

*Sound and other mechanical motions dissipates into heat

*Certain chemical reactions (exothermic) produce heat , whereas some (endothermic) absorbs it.

*Change of the state of matter produces/absorbs heat without raising the temperature (latent heat of fusion , latent heat of vaporization)

 A: leftaroundabout gave an excellent explanation for the thermal conduction
of insulators. However, in the case of metals, a significant amount of
energy is carried by the excitations of electrons (the width of their
Fermi-Dirac distribution). The thermal conductivity is then related to
how far an excited electron can travel before being scattered, and is
therefore related to the electrical conductivity. In most metals, the
electrons will have a greater contribution to the thermal conductivity
than the phonons.
A: The analogy is a very good one, because heat transfer is in fact modelled by phonons, which you could also use to describe sound waves.
The crucial difference is that sound waves have a much longer wavelength (at least in the range of some millimetres) than thermal phonons (not more than a few orders of magnitude bigger than the atomic lattice scale). These small-wavelength phonons can easily scatter at any lattice impurities, while the sound waves need macroscopic pertubations (like air gaps in an insulated glazing) to do so.
A: I feel that when a physicist speak about Heat he/she has a flow of energy in mind. Suppose that you have a rod and that the two extrema are held at different temperatures. Then the Fourier law states that there must be a flow of Heat from the hotter end to the cooler. When a physicist speak, instead, of the molecular motion he/she is thinking to the internal energy of the body.
Now when molecules and atoms are involved, it more likely that we must enter into the quantum world. By the way, we can make some few heuristic semiclassical consideration, namely we can apply Boltzmann statistics to the quantum structure of atomic and molecular spectra. A body that is immersed in a certain environment will be in thermal equilibrium state. Atoms and molecules receives energy from the thermal bath, but they also radiate energy in such a way the the total balance is "no energy exchange", therefore no energy flow, i.e. no heat flow.
We must note though that when we deal with atomic or molecular excitation levels, we are considering relatively tiny amounts of energy. Take as a reference the binding energy of the electron in the hydrogen atom, this being roughly 13.6 eV. Sounds excitations involves way more energy than this and in this case you can forget that the body has a quantum nature. You can treat it as a continuum and apply the laws of classical mechanics, i.e. the theory of elasticity and forth.
A: Here is a numerical simulation of heat transfer in a thread which is only one atom thick, each number is an atom, 8 is an atom with 8 units of kinetic energy, and so on: 
First left side has more heat energy:
8        0        0        0
After one collision between neighbor atoms:
4        4        0        0
After one more collision between neighbor atoms:
4        2        2        0
After one more collision between neighbor atoms:
3        3        1        1
As you see, there is some heat traveling at speed of sound. At the beginning of the simulation, when there was a heat difference of 8 between neighbor atoms, large fraction of heat energy traveled at speed of sound. 
(Smell does not travel at speed of sound. If some smell was transferred when molecules touch, then smell would travel like heat energy)
