What a great find! I would try a Mr Clean Magic Eraser on them. I use them on my acrylic tub which you're not supposed to use any abrasives or harsh chemicals and they work really well.

I second the suggestion for the magic eraser. They really get out stains well.

Try a 25% bleach and water solution.
If the armchairs were outside the stains are probably organic in origin and should bleach out.
We use this to disinfect used legos (for the kids), it also makes the legos very "new".

I third the magic eraser. If that doesn't work, try a real eraser next (a white one--an orange pencil eraser can leave a mark worse than what you had.) If that doesn't work, try a steam cleaner (cautiously--I don't think the heat would be enough to affect the plastic, but try a little bit at the back first to be sure.) Distilled water is a much more effective cleaner than regular water. You could try rubbing the marks with distilled water on a q-tip. Also rubbing with spit on a qtip--digestive enzymes can be the very best thing to break down organic dirt.
I wouldn't try anything with a solvent. You can start out with a solvent on plastic thinking everything is fine, and then suddenly after ten minutes work, the plastic goes all funny, sometimes nowhere near where you've been working--it can travel down the polymer chains, I think.
If it's truly a stain, i.e. penetrated through the plastic, just give up, you're never going to pull it out. If it's on the surface and just stubborn, keep trying things. Good luck!

I'd go easy on the bleach with vinyl. It's impossible to get all of the bleach removed and it can make it brittle. I'd stick with the magic eraser and/or trying the baking soda and vinegar trick. It won't harm the vinyl but will need to be thoroughly removed afterword. If bleach is absolutely your last resort I'd use 1T in a quart of water.

I'd use caution with bleach, it can linger and stick to the vinyl causing it to get brittle. If it's the absolute last resort I'd go with 1T to a quart of water. I think the magic eraser is a great plan, or try baking soda and vinegar, just be sure to get the vinegar completely off afterward. Can also cause it to get brittle but not nearly so much as bleach.

we've had good luck with oxy clean cleaning concrete,decks, clothing and color fast textiles... spot check first.

How about one of those green scotchbrite pad? or a very fine steel wool? Maybe try barkeepers friend as well.

Try using softscrub but be careful about rubbing it too hard. Magic eraser is good too I've used a combo of softscrub and the magic eraser to get out a lot of stains.

baking powder and vinegar could be good. for organic stains, ecover stain remover is great (it mimics the enzymes found in saliva so is pretty natural for a biological cleaner). Sunshine might also work!

I can't believe that no one has suggested Simple Green. It works great on plastics and the like.

From when they first appeared (and were semi-within-reach), I have a bubble chair -- and magic erasers have kept it clean, taking care of anything that has landed on it. The chair continues to look good as new.

Goo Gone.

If you'd like to try a more natural solution first, make a paste of baking soda with dish detergent. Even better would be to make a paste with a type of enzyme cleaner, Bio-Kleen (in the green bottle) works great on the stains I've used it for. I found it at my Coop. There are a lot of other enzyme cleaners out there though.

Thanks for all of your suggestions! I read all of them and just got off the phone with someone at Kartell's showroom in NYC who was actually much more helpful than the first person I spoke with there. I am going to try Oxy Clean powder with a little bit of water to work into a paste and scrub with Magic Eraser. Will send some photos in when it's all done and let you know how it turns out.

@gettysgirl, @medusa12120: holy crap, no! Goo Gone melts plastic.

I vote Rubbing Alcohol.

Try greased lightning. Its a yellow spray bottle available from Home Depot. Our patio furniture cushions were coated in black funk from outside/trees. We couldn't get it off with just regular soap and water. Greased Lightning. Spray it on. Lightly brush it with a scrub brush, rinse off with a hose. Spot check of course first.

kartell makes a cleaner specifically for their products that i use on my spoon chair. works great.

Cleaned the chairs last night, they look great. @Newmama08 I had no idea Kartell made a special cleaner, one would have thought when I called them they would have suggested it? Anyway, I did end up spraying the chairs with a garden hose, while they were wet sprinkled Oxy Clean powder on them, added a bit more water and scrubbed the chairs with the Oxy Clean paste and a Magic Eraser. Worked like a charm! They now look brand new, not bad for $50 each and some elbow grease.

Now I just need suggestions on how to get my neighbor (who's completely insane) to remove that hideous stained glass window from her window sill.