Air carrier crew reported while in cruise and deviating for weather they encountered an area of severe mountain wave resulting in aircraft over speed and the need for an immediate descent. Air crew descended and exited the area; flight deck conditions returned to normal and the flight landed safely at its destination.

Narrative

En route from ZZZ to MCI; ATC was calling mountain wave and turbulence ahead around the CIM waypoint. After a discussion on the flight deck and with ATC; we took a recommended southern course over ACH way point to avoid area of moderate turbulence and wind shear. The flight remained smooth until reaching ACH way point and starting to turn NE. We encountered an area of severe mountain wave; ultimately gaining +30 KIAS of airspeed and sounding aircraft aural over Speed Warning. We initially tried to correct the over speed by reducing power but the airspeed continued to increase to the clacker and beyond. I also tried to used the speed brake but the airspeed continued to increase. At this point; I disengaged the autopilot and auto throttles and the First Officer reported the condition to ATC. We also reported immediate descent. We continued descent to FL310 before the flight conditions improved and cockpit indications were normal. As we descended the VSI and airspeed continued to be erratic. ATC gave a vector to deconflict with traffic. After reaching FL310; the Flight Attendants reported no issues in the cabin; we contacted Dispatch to report the flight conditions and our flight deck indications were normal; throughout the remainder of the flight. We landed in Kansas City; Maintenance did required inspection and found no aircraft damage.Suggestion: We could have slowed airspeed as we approached the area of reported mountain wave. ATC could have given us a lower altitude over the area. The reports from aircraft ahead would have been helpful but we were one of the first flights that day. We assumed the south deviation would keep clear of the moderate/severe forecast area.