No you don't have to take apart the dash, as there is no real need to take the heater core out of the car.
Look at this picture:
The two longest/largest hoses leading out of the heater core, lead to your engine. So, disconnect the sides of those hoses that are nearest the two "E-15" labels. Then, with your car set to full heat (inside the cabin, since it controls the valve leading coolant into the heater core), push water through the valve side of the heater core hoses (top left in that picture). You should get fluid out of the top right hose.