How to clean inside of radiator removed from car?

My spec Miata engine blew chunks out the bottom, and what was left of the oil and water (not coolant) mixed. 
 

Replaced the engine and cleaned everything I could, new hoses, but old radiator and heater core.

I think I'll bypass the heater core so I don't contaminate the cooling system, but how can I clean the inside of the radiator without spending a fortune? If I put it on as is it will just contaminate the rest of the system.

Edit: do you think an engine shop could clean it?

Your degreaser of choice and lots of water running reverse flow will get the worst of it out.

A radiator shop should be able to do that. Should be under $50 depending upon local col. Beyond that, a few good soaks with your garden hose should be good enough 

Garden hose method. Try to find a combination of tubes and hoses for a good seal. Forward, reverse, forward, reverse, until it flows out clear. Do the same with the heater core. If stuff got caught in the radiator, it's probably in the heater core too. 

It's no big deal. It takes months/years of having a heavy oil saturation in the coolant to cause hose issues.  Oil rises to the top of the cooling system, it will all work its way out to the overflow tank.  Until then, it really isn't hurting anything being in there.

You can run water and dish soap in the cooling system if you really want.  I believe GM actually called out Cascade but anything soapy or detergenty will do.

Thanks for all the suggestions! I used a garden hose and some soaps, and sloshed it all around (multiple times), and then ran the hose through both ends for about 15 min each (crystal clear water at the end)

But.....I think I exposed some leaky areas, so I'll test that. I found a video of a guy getting expandable freeze plugs for the ends, and renting a pressure test kit so I'll do that tonight. Fun!

On the race engines we usually just install the regular frost plugs and then epoxy them in place.  Of course you have to take care to completely drain the coolant if the car will ever be stored in an area that might drop below freezing temp.

Update: the expandable freeze plug idea worked! Put them in the inlet and outlet, rented the pressure test, and no leaks! Only cost $10 ($350 if you include the rental....they really want their tools back)