I've used Murphy's oil soap for cleaning wood, but if the finish is tung oil or danish oil, I'd hesitate to recommend it. It's usually for polyurethane or varnish finishes.
What kind of wood are you trying to clean, how fine is the finish, rough or sanded well, these are questions that might help us to get a better idea of what you're dealing with. Some woods are very dense or hard and won't soak up moisture easily whereas something like pine is very soft wood and will soak up moisture or water more easily . You may want to try to post this question on the wood working forum, see if you can get any answers there. When cleaning large areas like walls and ceilings, I've used a mop with a flat head, you can wrap it with whatever material you need to use if the sponge isn't suitable, and thus cover more area. good luck and keep us posted. ;)

Carolsis, Thanks for the input. As far as I now it's white Pine. Full log construction and a smooth surface. I was thinking Murphy's and I've done spot cleaning without any issues. I also wonder if there's anything to put on after cleaning to help shine it up. Good idea about Posting to woodworking. Wish there was a log home forum.