A B777 flight crew on an international leg reported deviating from assigned altitude because of turbulence and performance issues.

Narrative

We were at FL388 or 11600 meters when we entered an area of light chop. We turned on the seatbelt sign; shortly after it went to moderate turbulence and we sat the Flight Attendant's. Nothing was painting on the weather radar and no towering cumulus were visible. Speed began fluctuating. Speed loss then became excessive and I began a descent turning off the autopilot and the other Relief Officer requested an emergency descent clearance from Control due to moderate heavy turbulence; but didn't feel it was severe. Control cleared us to 9800 meters. In the descent we had a momentary stick shaker. We then regained speed in the descent with a nose down attitude but the aircraft began climbing in an updraft. We requested and received clearance to climb to 11600 meters. When reaching the altitude we had exited the area of turbulence and were leveled off by [our next reporting point]. Then we reengaged the autopilot and checked with the Flight Attendants and passengers. There were no injuries to either. The Captain and flying First Officer were on break at the time. The Captain did a check around the airplane and reported two carts had tipped over in the aft galley but again no injuries had occurred. We then notified Dispatch of the area of turbulence.