I wouldn't use grease on the chain, it's likely too thick. There are a zillian specialty chain lubes for bicycles and motorcycles, I would think one of the "dry" wax-type lubes would work well, and resist getting gunked up with sawdust. Your local bike shop should have lots of choices.
If you don't want to use a specialty lube, Tri-flow works well, and stays fairly clean. Wipe the chain down after applying the lube, whatever is on the outer plates isn't contributing to lubricating the moving parts and will just collect dust. I like the liquid, drip bottle version of Tri-Flow, but it also comes in an aerosol can.