Hi, my question is:

How do you remove the sticky dust that accumulates on the fret board?

The dust what I'm talking about is that one which appears below the strings when you don't clean your guitar.

Glad you can help me.
There are many methods. But I suggest you just bite the bullet and buy some fretboard cleaner and conditioner.
Fast-fret usually does the job pretty well, it's just mineral oil really. It's cheap and it lasts a long time too!