I would go with grey walls, it works quite well with yellowish tile like that and white cabinets. Then incorporate some yellow, grey and white accessoires. I think this bathroom makeover is a good example: http://www.apartmenttherapy.com/sharons-bathroom-before-after-120622
The maroon tile is unfortunate, maybe its best to just ignore it. Or you could think about it if you want to risk to paint it black (depends on your landlord).

If your landlord allows you to you could paint the cupboards. Or for a more playful temporary idea, you could find some wall paper or even some fabric that has the colors you like and put it in the inside middle panel of the cupboards? Even some scrapbook paper might work ;)

I live in a 1920s house as well and have similar tiles in my dining room. I LOVE them. I consider them a cool old reminder of a different era and would NEVER remove them. But I don't try to decorate around them. I sort of treat them as I would a Persian rug: attempt to blend, but never to match. So my suggestion would be to pretend as if the tiles aren't even there and decorate as you would if you had an all white kitchen (which it looks like you essentially do). The tiles, in the end, will just be a cool accent, sort of like a great cocktail ring. You're really lucky to have them!

How about door handles/knobs to match one of the colours?

Well, first off, yellow can work well with grey. You can find a nice soft grey color for the walls if you are allowed to paint, and if you have any windows, you can look for a fabric that has similar tones as the tiles and your accessory colors to tie things in. I own a house built in the 50s. Both bathrooms are tiled up the walls. One is peachy-pink and the other is yellow. I have a very soft grey in the yellow bathroom and a bright robins egg blue in the peach bathroom. To tie in the colors.

Other than pulling in a pattern (if you prefer a more streamlined approach) You can always do some simple towels or curtains with a burgundy accent on them. A quick search of the color combination pulled up the third picture in this post. http://littlenudge.wordpress.com/2012/02/27/unexpected-colour-schemes-floral-inspiration/ There are a few color scheme ideas there that could give you some inspiration. They are a little brighter than your typical colors, but you could learn to love it.

I have the same tile and cabinets but my apartment is from 1959. Interesting.

The tile blends in once you have all your stuff in. It won't look weird. I second Creative License comments in regards to it

If you can paint the cabinets, paint them gray.

oops, first paragraph of my first comment should have said "To tie in the colors, I have a shower curtain with the pink, blue, and some other bright colors to pull it all together."

That mustard yellow can muddy so easily that I would leave it surrounded by bright white, but pick up those two colors in an accent or two. A rug, a fabric shade or curtain, upholstery on chairs or a banquette, a large ceramic piece, or artwork on the walls. But only a couple of these at most.

I agree with Creativelicense. I have similar tiles in my kitchen but they are yellow and baby blue. Instead of trying to find a way to decorate with them I ended up decorating my kitchen how I wanted and ignored the tile. It worked! The tiles tend to just blend in.

I'll add my voice to the gray chorus -- a warm gray could definitely be cozy on its own and striking against those colors.

Keep the yellow, go with a warm gray.

Add Orla Kiely textiles. They have the same set of colors and are modern; they'll tie it all together.

I have lived in kitchens with colored tiles most of my life (my parents' house and then several apartments/cottages) and find that they don't need to be matched. My kitchen decorating style has definitely been influenced by those vintage kitchens, but I just make sure my dishtowels are in a color that doesn't clash and then choose vases and other decorative items that are neutral or that seem to flow with the tile in some way. Whatever you do, don't try to pain the tile! I bet you'll end up loving it.

I'm not seeing a way to incorporate industrial style into this, but your preference for midcentury ought to blend in easily. My first thought, though, was Tuscan. There's many, many Tuscan-themed wallpapers and coordinating fabrics (and towels and aprons ...) that tie together the golden and russet tile colors as well as any gray you might introduce. A lot of these patterns tend to features olives and olive sprigs. Just a thought.

For those of you who have the same in your kitchens, do you live in LA? I have the EXACT same tiles in yellow and maroon (and also despise them) and have seen them again and again around the city. Was there a surplus of yellow and maroon tiles in the 1920-30s?

Anyway, I feel your pain. We're working on pouring our own concrete counters, but in the mean time have more or less covered them with several butcher blocks. They're not permanent, but they're handy and a little more attractive.

Hail to the Redskins!

I agree with Creative License - blend, but don't match. The tiles are the statement, in and of themselves. I am reminded of a project I read about recently, by a pair of Barcelona-based artists, who value the quirks of our interior living spaces, in a world that is ever attempting to iron out imperfection: http://www.smart-urban-stage.com/blog/future-of-the-city/new-ways-of-sanctuary/ I think you can look at these tiles in the same way - they are a daily reminder of a bygone age, a layer of history in your own home.