Are you replacing all the flooring or just the checkerboard?

I would look at replacing the black and white with a hardwood area that is set off from the existing hardwood by a contrasting border. That allows you to have a different style hardwood flooring in one area than adjacent areas. A hardwood floor pro can help with design. Take a red cherry drawer or cabinet door to the flooring store, along with photos of your other hardwood. You will need to bring home samples and there will be a lot of back and forth while you explore options.

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Here's a search link: http://www.houzz.com/photos/query/hardwood-floor-border/nqrwns

By the way, repainting the white risers in the creamy wall color you already have would reduce the contrast and allow you more flexibility in the flooring choices.

A wood in a natural finish (light gold) will work. White Oak, maple or birch will do very well with the island.

Ideally the stair treads would work with the floor that it is touching/at the landing. You might want to leave a small amount of this tile in place at the door entrance/landing area. That little bit of black/white tile will allow you to leave the black/white stairs in place. They would each give the other a reason for being in the home.

The rest of the tile would be removed and the lower level finished in a wood colour of your choice.