you might try some of the vegetable based oils like
http://www.lubegard.com/industrial/chainsaw.html. . They MAY not stain as badly. I use canola oil straight up on the bronze bushings on the water powered sawmill I run, because I didn't like the idea of the petroleum based lube going down the stream.
However I use Kero on the blade when I see some pitch building up. While it stains the wood when new, I don't remember seeing as a stain/streaks on the air dried wood.
Bob