Your style looks very French Country to me. I would look for something that has the appearance of being hand-made or stone - something with a lot of variation, but probably a simple pattern. Here are some examples:

Farmers - Showcasing projects built and designed by The Maker Designer Kitchens · More Info

Corian® Quartz - Imperial · More Info

Mediterranean Kitchen · More Info

Traditional Kitchen · More Info

I do not like rough stone anywhere near a stove and FYI the backsplash IMO needs to go with everything in the space not just the cabinets and for sure needs to go with the counter

You've already got a range hood that is a powerful visual A-player (like the star in a movie). The cabinets are high end - basically B-players or the supporting artists. IMO, the backsplash should be the C-players or chorus line - a simple smooth tile in a simple design like running bond or square grid. This will allow the stars to shine without competition. As Patricia says, raw stone or anything rough is not a good idea, but you could use a tile with a crackle or a bit of unevennness to match the mood of the country hood.

Below are two drastically different backsplash ideas. You could look into getting one custom made, or try something on the simple side to make the hood the star.

Kitchen Dreams: Custom Backsplash · More Info

Kitchen Dreams: Wood Comfort · More Info