Well, if the wood is unfinished and unstainek, completely natural, I wood try with lemon oil. I wouldnt try this in a BSB because i dont know if the stain would be partially removed. Of course, you can restain a BSB wich is pretty easy, but I dont know of I would risk it unless you are able to restain it in case you screw it up.
I owned 2 unfinished guitars (a PC3 and a BSB). I clean the PC3 with lemon oil and it vastly improved. I didn have to clean the BSB, but I did have to restain it.
May be Rich has a better tip about this.