Can I wash the cloth I use to polish and clean my wood furniture when I use Murphy's Oil Soap?

I have lots of antiques, and I’ve always used Old English to polish them. But on a remodeling show they showed how wood soap really, really cleans, so I started using Murphy oil soap, and I like the results MUCH better. They just look and feel cleaner. The thing with Old English though, is that I just used the same oil cloth for years. With the wood soap, however, I have the urge to wash the wash cloth after three or four uses but…I don’t know what it might do to the rest of the load in the washer. Anyone know? Any ideas on how to wash them, or if I even need to?