Bicycle shops sell chain lubricants formulated for the same condition (less attractive to dirt). They generally contain more wax than oil, and the best treatment methods rely on application, rub down, then allowing plenty of time to dry before use (as Dave suggests). They come in a handy squeeze bottle, like suntan lotion.
The other option I know of is Corrosion-X
family of products. That's what I use on leveling rams, and for general purpose machine oil substitute. Again, apply, rub, wipe down to nearly dry.