If you are concerned about it looking dated, it seems a red flag that you perhaps don't really love the look enough to do it for the sake of your personal aesthetic. I say paint them the same color as your walls. If you need a punch of black introduce it through antique pieces.

Black doors seldom look as though they belong in a home.

I don't think I would do the black. Doors in most houses are put where they are needed to be functionally, so some are closer to each other than others in hallways, some doors right next to each other will be different widths, you don't necessarily want to accent every closet door. I think there are a lot of things, potentially, that you don't want to emphasize.