When is temperature not a measure of the average kinetic energy of the particles in a substance? I had always thought that temperature of a substance was a measure of the average kinetic energy of the particles in that substance:
$E_k = (3/2) k_bT $ 
where $E_k$ is the average kinetic energy of a molecule, $k_b$ Boltzmann's constant, and $T$ the temperature. (I'm not sure of the 3/2 coefficient.) Then I heard from several folks that this is a simplistic notion, not strictly true, but they didn't explained what they thought was flawed with this idea. I'd like to know what (if anything) is objectionable about this idea? Is it that the system must be macroscopically at rest? Is it that it ignores the quantum mechanically required motion of particles that persists at low temperatures? When is it not valid?
 A: There are a number of ways of defining temperature, for example using:
$$ {\partial S \over \partial E} = {1\over T} $$
This definition is the basis of negative temperatures. This is a case where the temperature is not a measure of average kinetic energy.
A: The expression your wrote down for the energy is the expression for the ensemble average kinetic energy of a monatomic ideal gas.  Therefore, we see that for this system, the average energy of the system is simply proportional to the temperature.
For a general statistical mechanical system, however, it might not even make sense to talk about the "average kinetic energy" of the system.  For example, take a quantum system consisting of a single spin $1/2$ particle interacting with a magnetic field.  It is possible to define a temperature for this system when it is in contact with a heat bath even if the particle is not moving around.  In such cases, one appeals to more general definitions of temperature such as that to which John Rennie refers in his answer.
A: 
temperature of a substance was a measure of the average kinetic energy of the particles in that substance

This is only true for a monatomic gas. Even without quantum mechanics, a classical diatomic gas has three more degress of freedom than a monatomic gas—two rotational and one vibrational—for a total of 6. The equipartition theorem tells us that the kinetic energy will be divided equally among these. Thus the kinetic energy of the gas particles in this case represents only 1/2 of the energy represented by the temperature.
