Here is what I do.
I take 00 steel wool and dip some rubing alcool to it.
then I rub it up and down the fingerboard to remove the grime from my fingers sweat spots etc...
(make sure no alcool goes down the side of the neck and come in contact with the finish it might ruin your finish)
When the alcool has dried I dip a peice of scott towel with lemon oil and rub it on the fingerboard you dont need much and you will see that the wood
absorbs it a bit too. Just make sure you dont get any on the face. if you have duct tape or something put it on the edges of the face where the
fingerboard gets close to it so it protects it from geting oil on it.
If you have a pick gard that is also unfinished it may start to develop cracks if you dont protect it with lemon oil. I noticed that on one of my ouds
and I trimed the outsides of the pickgard with tape and aplied a bit of lemon oil. it made it much better.