Flight Instructor on training flight reported engine running rough in cruise and returned to the departure airport.

Narrative

Training flight for a CFI student; the flight started normal. All preflight checks were completed; and the weather was checked before the flight. On taxi I (Instructor) noticed the CHT (Cylinder head temperatures) rise quickly but nothing out of the normal operation area. Upon climb; the CHTs went up to 275 (start of red caution temp is 260). Put the plane into cruise climb and pulled the power back from climb setting of 5000 RPM to 4400 RPM. CHT's dropped back into the green and we continued the climb. Upon reaching our ALT of 2000 ft. MSL we introduced cruise power and the engine immediately started running rough; vibration was felt in the stick of the plane and on the avionics panel. An engine 'grinding' sound was heard from the engine and significant engine roughness was not only felt but audible. Contacted ZZZ Tower; turned back to the field; and [requested priority handling] with the Tower. Kept the power setting where it was (4400 RPM) and was cleared to land number one. Landed and taxied with no further problem.