How can I remove grease and dirt stains from unfinished wood?
Here's a few tricks I've learned from cleaning up a few old wooden handplanes, where I wanted them to look nice without taking all the patina off:


give it a rub-down with oil, either curing (e.g. linseed oil), or non-curing (e.g. mineral oil, camellia oil). If you use a curing oil, it's more permanent. This may even out the stains a little bit without removing them entirely. It also can bring out the chatoyance in the wood and act as a minimal finish. If you pick a darker colored oil, it may disguise the (presumably dark) grease stains better. (NOTE: I'm talking about oil, not finishing products that call themselves oil. For example, danish oil usually contains polyurethane, and "tung oil" finishes could be anything.)
try cleaning with mineral spirits (paint thinner) . This will soak in and may partially dissolve the old grease and even it out a bit. Like adding oil, this will not remove it completely, but it's pretty harmless to try (as long as you wear a mask and stay away from open flame).
If you want to be more aggressive, try using some fine (like 600 grit) sandpaper by hand. This will remove some patina, and may require you to sand the whole thing--so think carefully before proceeding. But if you go lightly, you may be able to improve one spot and fade it into the surrounding area. If it's too slow, you can always use coarser paper.