Generally, if the fretboard is just dirty with every day playing, I'll use just water in a spray bottle and cloth. For particularly dirty parts you ball up the cloth into a knuckle ball.
For really hardened dirt, use a little naptha on a Q-tip to get it wet & loosen it, then use something like a plastic radius gauge to gently scrape it off.
Keep the plastic even on the fretboard.

0000 steel wool can also be used gently on the fret/fingerboard joints.
When you're finished, use a fresh ball of wool to polish the entire fingerboard in in one direction along with the grain.

(don't use steel wool on a maple fretboard)

To finish off, if your fingerboard is dry (usually a white hazy look), use a dab of lemon oil to condition it.

Like with anything though, don't overdo it. You don't need to use steel wool or lemon oil every time you clean the fretboard.
Only do what's necessary.
There is actually a balance between the right amount of natural oil from your fingers and a dry fretboard.