Mucky stuff in raditator

The greasy stuff could be the signs of a head gasket begining to fail. The oil pressure and compression in the cyclinders is always higher then in the cooling system. Small amounts of oil, gas or exhaults fumes could be being force thru the failure and causeing greasy stuff. Usually looks milky brown lite grease.

Then it could be a sealant Bar's Leak used to make a fix by some on minor radiator core leaks.

There is a chemical test to determine weather there is exhaust gases present in your cooling system.

Did it start suddenly, or did it grow over a longer time? Did the highly trained personnel at the oil change place start pouring motor oil in your cooling system?

Try flushing your cooling system. you will need a non foaming detergent, and lots of patience. You should be able to flush it yourself, or pay to have your cooing system chemically tested. Don't let someone convince you to remove the radiator to flush it out. There is just as much mucky stuff in your engine as in your radiator. Good luck.