B767-300 pilot reported encountering wake turbulence in cruise flight from a preceding heavy jet that made it difficult to maintain airspeed and altitude. Flight crew asked for and received clearance to a lower altitude.

Narrative

We departed DXB at max gross T/O weight 407.4lbs the First Officer was the Pilot Flying and I was the Pilot Monitoring. We climbed at 320/.78 to FL240 per the flight plan. When we switched to Bahrain Control they asked if we could accept FL300. I looked at the VNAV page in the FMC it said max FL was FL32.1. So I accepted the climb clearance. Climbing out of FL290 we started to encounter turbulence. As we leveled at FL300 the aircraft started to accelerate to Mach .82 per the flight plan as soon as we leveled off the turbulence be came continuous the aircraft tried to overspeed. The Pilot Flying opened the speed window to .78 and manually pulled the thrust levers back to midrange maybe slightly more. Then the turbulence's increased to moderate. The aircraft was pitching up and down +/- 3 or 4 degree's and the wings rolling +/- 20 degrees trying to stay leveled at FL300. Then; the autopilot disengaged we received the master warning for autopilot disengage. The aircraft had started to descend. The Pilot Flying then selected the vertical speed as the aircraft had started to slow down and descend. When the vertical speed button was selected the aircraft had a Ivvi of 800ft to 900ft down due to the turbulence. I the Pilot Monitoring saw what had happened and re-selected the C auto pilot and immediately selected flight level change. The lowest I saw the Altimeter read was 29;700ft. Bahrain Control said he showed we had descended enough to get an Altitude warning. I called Bahrain and reported moderate turbulence and at this point I had started to suspect it might have been wake turbulence from a Heavy Jumbo that had departed DXB ahead of us. We were in continuous turbulence so we requested a descent to flight level 280 with the ride improving in the descent. The rest of the flight was uneventful.