Those floors are gorgeous! Why are you wanting to change them? Is it because they are similar in color to the cabinets? There is an overall monotone feel to it, but that is more because of the cabinets in my opinion, there is a lot of them! They are beautiful though so I could see why you wouldn't want to paint them, you could stain THEM darker but that would be major work, harder than the floor anyway. I'm not sure any drastically different color on the floors would look good with them though. All I could think is to suggest is to stay with the same reddish tone but go a shade darker or lighter than the cabinets to get contrast but not clash with them.

And here's a totally off the wall idea, I'm not an interior designer so take with a grain of salt. but if the toekicks of the cabinets were painted another color it might help it not feel like the same color running from the floor up to the cabinets.

These are what I found in a quick search, maybe not the best examples but just to give you an idea of what I mean.

My favorite stain is a medium toned one--Special Walnut. It's not too light or dark, and has a neutral brown color, no obvious red/orange/yellow hues. I think it would look nice with cherry cabinets. But it's hard to really discern to color of your cabinets from the photo, so you need to try it out in the room and see it for yourself with your lighting, both natural and artificial.

I would not mess with the floors. Get some hardware on the cabinets, will help break up the monotony and also be functional.

Removing the 4" backsplash and replacing with a full backsplash is something I'd also think about.

Here is how your kitchen might appear with a slightly darker stained floor

I would 86 the hanging fixture. It's distracting and dated.

Leave the floors the same, paint the island and walls, add backsplash.

I'm in the same situation as you currently. We attempted to go mid-tone warm gray with our lighter cabinets - stain look great during the daylight but at night it showed a blue-ish tone. As a result - finalized on no stain on red oak. Just sealer and Loba 2K Supra finish. Lesson learned - Simple and natural.

Dark floors show scratches and dust much more than light floors - daily Swiffering needed. Partly because of this, many are trying to refinish their floors back to light, but that is difficult to do. The dark stain gets deep into the grain and into the end grain of the wood - very difficult to sand totally out. I saw an attempt to bring dark floors back to light at a remodelers' parade and it was not very pretty.

It looks like the floors have been stained/finished to match the cabinets. If you must refinish them (and I would ONLY do this if they ABSOLUTELY need it) I would work with a water based polyurethane. Just as dcarendt has done, going NATURAL with a water based finish will offer a degree of separation which would be JUST ENOUGH to offer some visual interest.

Personally I think the COUNTER TOPS need changing more than the floors. My first choice would be a CRISP white (no gray) or a soap stone/slate counter top would be EXTREMELY handsome.

You need some CONTRAST in the kitchen. Right now you have mid-tone sitting on mid-tone sitting on mid-tone. You have the lightest version of "brown blahs" I have yet encountered.

Changing the counter tops would be faster and probably cheaper than changing the finish of the wood.

If you do change countertops, consider moving the stovetop to the countertop to the right of the sink. Remove two or all of the 4 upper cabinets for a vent hood. Venting a gas cooktop on an island is difficult. Cooking on an island can be difficult too.