depends on what kind of finish you want. if you want a stain, than mdf will not work for you. and i'd be leery about using mdf anyway, it doesnt hold screws or nails well and you get some swelling with any exposure to moisture. to avoid that you'd want to paint/seal all six sides of any panel.
If you're looking for something with not much visible grain, I'd just go with a birch or maple plywood. The grain will be visible, but since its not open grain like oak it wont be as prominent, especially if you're painting it.