I tried armour all but it left grease shines and plain water doesnt work great.

Try a medium cut polish on a microfiber/terry cloth applicator for some cleaning AND scratch removal.

You can try Meguiar's plastic polish/cleaner (cleans up the small scratches and adds shine) and then finish with any automotive wax (more shine).

What car is this on? That way we can tell if it was real wood or plastic wood.

99 STS, zebrano wood.

I would just use a soft cloth, microfiber maybe, and some Pledge wood cleaner.

Unless I'm way off base, the REAL wood in these cars is coated with a poly-something finish anyway - it's not 'bare' wood like a dining room table or something. So you really want something that will clean the grime off the 'plastic', if I may call it that, and then apply a bit of protective gloss/shield. I've found that any good all around cleaner APC like Simple Green or whatnot is good for cleaning, and then I coat the "wood" in my Eldo with Duragloss Aquawax. But any halfway decent protective coating should do - just remember you're not waxing bare wood.

I know this is not bare wood, but the wood is treated very similarly to most furniture. I was just wondering because the wood in my old '88 FWD Fleetwood was real wood without any plastic coating so I had to use Murphy's Oil Soap to clean it.