Not to contradict Steve, but do not take it to a radiator shop. They "boil" out the radiator/cooler and that will leave debris/water in coolers we use; it is okay for a radiator with big tubes they "rod" out, but not closed passages.
The radiator shop I use plugs the trans cooler when working on the radiator portion. When they are done with the radiator, they use lacquer thinner to flush the trans cooler.
I do the same thing at my shop; I filler the cooler with lacquer thinner, cap the outlets, agitate the cooler and then let it sit several hours and/or over night. I do this process several times until the thinner leaving the cooler is clear/clean. The thinner disolves the oils and releases any particulate matter from the tight tolerances of an oil cooler.
The best deal on lacquer thinner is to buy the 5 gallon, "wash thinner" at the auto paint store. This thinner is a lower grade than used for paint, and is used for the cleaning of the painting equipment.
Good luck,
Randy - RPM