A lot of guys will probably beat me down for this, but I use my gun rags to clean my carbon and damascus blades (not my kitchen blades, those are indeed mineral oil.)
There is a product called "rust prufe" you can get from most local gun shops that really, really does the job. Leaves a very thin film, and after a day, you won't even notice the film. I live in Florida, so my steels see some wickedly awful conditions for corrosion. My truck gun rusted the first day i left it overnight, but ever since rust prufe once every couple months, nothing.
Comes already applied to a rag, and comes in a cute little jar. Smells heavenly.