I've read that Gojo is a good water-based cleanser for sewing machines. I've never used it, though. I've used damp sponges for really caked-on dirt: I moisten the layer of dirt enough that it softens so I can gently wipe it off with as little rubbing as possible. But I imagine that sewing machine oil would work in exactly the same way.
TR3 is one I use and I've found it does a good job on all kinds of machines and their surfaces.