First, prep your piece. I've discovered this new cleaning product that I have been raving about lately! It's called sugar soap and it's made by Frenchic®. This is an all natural cleaner that will get up dirt, built-up grime, even nicotine stains! You just wipe down the piece using paper towels or a rag! If the piece is super dirty, I recommend using your cleaner plus extra fine 0000 steel wool to scrub the piece.
My piece was relatively clean compared to some others I've worked on. That dirty paper towel pictured below is after wiping just the top of the dresser! It picked up quite a bit of grime that I didn't even know was there!