Make a Sandpaper Dowel
Some gentle scraping with a folded fine-grit piece of sandpaper will rub out dirt and stains on tile grout. But if you have thin tile grouting like most kitchen backsplashes, these areas can be hard and strenuous to reach.
To combat this issue, sand down one side of a dowel on a clean power sander, and then the other so the tip is slightly thinner than your grout line.
Once done, simply cut and tape a strip of sandpaper to the end of the sanded-down dowel. Now lightly sand your stained and dirty areas of tile grout.
If you are worried about scratching glazed surfaces like tile, simply cover them with painters tape, and remove when done.