Squirt enough baby oil onto a lint-free cloth to dampen it.
Rub the damp cloth onto the stainless steel surface, using large circular buffing movements. Repeat until the entire backsplash has been cleaned. This will remove dirt and residue without leaving streaks behind.
Buff the backsplash dry with a second, clean lint-free cloth. This will remove any remaining oil or residue, and restore the natural shine of the steel surface.