If you don't already have some, I'd purchase a spray can of multi (gun) purpose Ballistol purchased direct from ballistol.com (the World's Most Useful & Environmentally Friendly Lubricant), also Midway USA or Brownell's. It lubricates and protects firearms, lock, stock & barrel. It not only cleans, preserves and protects metal & wood, but restores and revitalizes leather and plastic. The nice thing is that if any of the Ballistol remains in the pores of the metal in the grips, it won't cause rust and doesn't harden or gum up. Spray it on the grips and let set a few minutes to do its thing, than use the toothbrush to work it in real good. If you install them on your 586, the next time you do a good cleaning on that gun, you won't need to remove the grips.
I'm sure the other recommendations will work, but if you haven't tried Ballistol, I'd highly recommend it for all your firearms, or anything that can use lubrication, your vehicles or things in and around the house. Have fun!
the Shadow