Some laundromats have one machine dedicated to greasy laundry.
Still, remember that wherever you wash the rags, the oils and toxic chemicals go down the drain - either into a public sewer or a private septic system, neither of which is equipped to remove those chemicals from the water.
I just use the heavy duty paper towels, try to reuse them as many times as I can, then pitch them. At least the landfill is supposed to isolate their leachate from the aquifer.
If you have a friend who works at a car or motorcycle shop, maybe you could arrange to get a couple of shop rags a week through their towel service. Although there was a highly publicized decision a couple of years ago that allows towel services to put a lot more effluent into the water system than they need to...
Speed comes from what you put behind you. - Judi Ketteler