The confusion in all the answers stated above is that somehow people are not making clear distinction between heat capacity, internal energy, temperature and kinetic energy. The point of the original statement that "...rotational kinetic energy of gas molecules stores heat energy in a way that increases heat capacity, [but] this energy does not contribute to temperature" is that there is energy which does not get realized as a thermodynamic temperature.
In the ideal gas the state variable temperature is only definable in a state of equilibrium. As such we need to expect that the gas is in equilibrium with the walls of its container. Of course this is an ensemble average condition, but that's the game of thermodynamics.
The reason we need to consider equilibrium is two-fold: First, state variables such as temperature, pressure, etc have no meaning outside of equilibrium and second, the microscopic model on which the definition of temperature is built assumes elastic collisions between molecules and containing vessel.
This is where the translational kinetic energy comes in. When a molecule collides with a wall of the vessel (thereby producing pressure on the walls), that pressure is a consequence of momentum transfer. It is not and cannot be a consequence of angular momentum transfer. Furthermore, there is no momentum in a vibrational mode of a molecule since by definition they are consequences of internal forces within a molecule which can never change the momentum of a system.
So the bottom line is that an ideal gas can only produce pressure against a vessel due linear momentum transfer which only depends on a molecule's translational kinetic energy. In this case it will satisfy the ideal gas law with a temperature T that will never depend on rotational, vibrational or any other type of motion or energy.
To suggest that the temperature of a polyatomic gas molecule is harder to change is irrelevant. That's heat capacity, and of course by the equipartition theorem it will be larger than for a monatomic gas. Regardless, at the same temperature both the monatomic gas and polyatomic one will have the same mean translational kinetic energy, will produce the same pressure on the walls of a vessel and will satisfy the same ideal gas law.