2002 Lincoln Continental Cooling/Circulation Problem

My 2002 Lincoln Continental has a cooling problem. When I stop at a red light, the heat begins to go cool. Once I accelerate, the heat gets hot again. There are times however, that it will go cool and not get hot again and the temperature guage rises to the red. Also, if I start the car up and let it idle, it will overheat. I took this to my mechanic and he replaced the thermostat and confirmed that the radiator was not plugged. In additon, he confirmed that the water pump appeared to be in working order. He also performed a chemical test and it came back clear. He has bled the system for air pockets and it still continues to get hot. He has all but given up on this problem at this point. What could be the problem?