I've had some success with flushing out the radiator and engine with a garden hose shoved into the upper radiator hose and removing the thermostat from the engine and turning the hose on all the way to reverse flush your coolant system. Wrap some rags around the hose to seal it to the hose.
After that, I've run water and a radiator flush chemical through the engine and driven it for a while to get it all the way up to temp, drain it out then do the garden hose trick again. After that refill the system with 50/50 water antifreeze. It isn't perfect but it'll get most of the gunk out of there for a minimal amount of money.
Some radiator shops will flush out your radiator too, but that won't get the gunk out of your engine.