I like the mineral spirits, too. They will dry off. I will only add that, after they dry, if you have a linseed oil finish you may find the oil has been pulled a little from the surface with the dirt and then you'll want to rub a coat of fresh linseed oil in to return it to original condition. As mentioned earlier, if this finish is antique, even removing dirt can lower its value, so check on that first.
Nick