The pilot reported that light rain and trace clear air icing were forecast along his intended route of flight, and he encountered those conditions during climb out. As the airplane was climbing through 14,000 ft mean sea level (msl) in instrument meteorological conditions, he noticed the airspeed had decreased 10-15 knots. He checked the wings for ice and did not notice any accumulation but activated the pitot heat at that time as a precaution. After the pitot heat was activated the Primary Flight Display (PFD) and Multi-Function Display (MFD) displayed a red X and went black. Subsequently, the autopilot commanded the airplane to descend. The pilot reported that he was unable to read his standby instruments due to the violent shaking of the airplane during the descent. As the airplane emerged into VMC conditions, the airplane was in an unusual attitude. He disconnected the autopilot and was able to recover the airplane to a level attitude. At this time, the PFD and MFD operation returned. An air traffic controller reported to the pilot that he had lost about 5,000 ft in altitude and airspeed had increased over 200 kts. The pilot responded that his avionics were working again, and that the aircraft was operating normally. He continued with the flight and landed without further incident. Substantial damage was discovered to both wings following the flight. The airplane’s “Before Takeoff checklist” calls for the pitot heat to be activated for flight into icing conditions when visible moisture below +5° C, is anticipated or encountered. A Federal Aviation Administration inspector examined the airplane after the event and verified the pitot heat was operational. The circumstances of the accident are consistent with the pilot failing to activate the pitot heat in a timely manner, which allowed ice to accumulate on the pitot static system. The PFD, MFD, and autopilot subsequently malfunctioned and the pilot lost control of the airplane.