Do all adiabatic processes involve a change in temperature?
The first law of thermodynamics says,
$$\Delta U = Q - W$$

By definition, no heat is exchanged between the system and the surroundings in an adiabatic process. Therefore, the 1st law reduces to,

$$\Delta U = -W$$

In an adiabatic process, $PV^\gamma =const. $ has to be followed. If you change pressure, the above equation implies that volume changes. If the volume changes, there will be some work done which in turn will change the internal energy of the system. Therefore, the temperature changes.

You can also come to the same conclusion using ideal gas equation. 

We know the ideal gas equation,

$$PV = nRT$$

If there should be no change in temperature, $PV$ should remain constant.

But in case of an adiabatic process, we know that
$$PV^\gamma =constant$$

So if you change one of those parameters, the product $PV$ will obviously change (so as to keep $PV^\gamma$ constant). Hence, temperature always changes in an adiabatic process.

*I have ignored ridiculously insane cases where matter can exit/enter the system through an adiabatic wall (which isn't possible). If you could discover a wall that allows matter to move across and does not allow heat to conduct through it, you deserve a nobel prize.

*Free expansion and mixing gases are two processes where the process is adiabatic and there is no temperature change.