Most dealers have a flush machine that was made just for that purpose. It uses a pump to circulate solvent through the cooler and lines. The same machine usually can flush the torque convertor, although that isn't done much any more since it isn't a good idea with lock up torque convertors.
Just using compressed air to blow out the lines and cooler won't do a satisfactory job.
So, no, I don't know a good shortcut for a home mechanic other than taking the radiator out and having a radiator shop flush it out for you. Compressed air should be satisfactory for the lines though.
