As much as I am not a fan of wood floor in a kitchen, though others are happy with it, I think in this case it would be the best solution, given the size and shape of the area. Use throw rugs in the doorways and under the sink to catch the worst of the wear and tear. And be sure to put plenty of coats of urethane on the wood if it's not prefinished.

What's your climate like?

I ask because if you live in a warm one, having tile everywhere might be desirable not only to you but to potential buyers as well. if so, you can choose anything you like, and might want to consider wood-look ceramic tiles. The one below measures 36" x 6" and has an antique/distressed look and optional decorative scribed tiles; it is from an Italian company, Fondovalle.

If you are in a cooler climate, you may want wood in the living rooms, and tile in your entry/mudroom as you've planned. You can still use wood-look ceramic in a herringbone pattern in the entry, in a similar tone to your wood floor to give you continuity of color without trying to lay fake wood next to the real thing. In this case I would opt for wood in the kitchen for a more uniform look.
{{!gwi}}

Of course, slate sounds gorgeous, and if you are going with two different floorings, I would keep tile in the entry/mudroom in a rectangular shape, and wood in the kitchen/living room. Large format slate tiles will look great, especially since the area you're tiling is small, and it'll give you few grout lines. If you choose slate, consider staining the floors a walnut color.

We have a similar situation in that we had maple floors in the kitchen, which we kept, but we added a mudroom "area" with quartzite (looks like slate) at the new patio door. This quartzite matches what we have in our "formal" entryway. We originally were going to do an arc, but we went with a rectangle and I'm so glad we did:

You are all so helpful. I think I am going to go with the wood. Like Anniedeighnaugh, I don't love wood in the kitchen, don't get me wrong, I don't hate it either. My ambivalence is probably why I am so torn about this decision. Aesthetically I do feel this space should be wood, as the transition otherwise is too harsh.
EAM I am definitely in a cold climate- Maine. So a large expanse of tile does also seem cold. Probably another factor in my inability to make a decision. I don't know if I can do the wood floors in walnut though, as the kitchen cabinets are going to be natural walnut. Right now the rest of the first floor, all open floor plan, is natural ash. When I refinish all the floors to get the new to sync up with the old I could stain, but I don't know. What does everyone think?
Leela, Your space is lovely and I think it is very similar to what I will end up with. I am not committed to the arc or to the slate and won't have to make those committments for a while thank goodness.

In Maine I'd leave stone in the entry, as leela4 and friends in Brunswick have done, but if you think all wood is the way to go, enjoy it and don't worry about the mud and snow! With natural walnut cabs, I'd go for natural ash - your floors look like they are already stained, or maybe it's just the image. Unstained ash and unstained walnut are beautiful together. I cut the cabinets in this image out of one of sparklekitty's old posts.

{{!gwi}}

Again, thanks for all your help today. My husband and I discussed it this evening(over margaritas) and we decided to go with the wood floor through the kitchen with some stone or tile right in the entry way and through the mudroom, laundry, powder room area.
I am relieved to have made another decision and I vow not to rethink it. Thanks for all your help, advice and support as always.

I'd carry the wood thru the kitchen so to unify the space so it doesn't look chopped up.
I live in a warm climate, and originally had tile in the kitchen. It was still cold in the cooler months, but worse, it was soooo hard to stand on. Hated it.

Have laminate now which was a huge improvement for comfort and keeping clean, but next remodel going with hardwood floors and carrying it thru the kitchen too because it's a somewhat open floor plan.