Q:

Is there a polite alternative to "No thanks, I'm full"?

English is not my native language, but when I was studying in the US, I was always trying to find an alternative to 

I'm full!

I felt that it was a very improper way to express that I have eaten enough, especially when I was invited for dinner. I remember having difficulties demonstrating my gratitude towards my hosts when using this expression.
Being full feels to me a bit like a car that has been filled with gasoline and gasoline is just used to continue working. That's not what I want to say... 
Is there a better word I can use? I haven't found one in the dictionary. There are, of course, long forms:

I have had enough [, thank you].

that I have learned to use, but I wonder if there is a short one.
Other Languages have rassasié (French), sazio (Italian) and satt (German). Should I use satiated?

Update: Thank you all for the overwhelming response! After 13 answers, I feel I got a better understanding of the phrase. I'm grateful for @Andrews answer, because he took the cultural aspect into consideration. This was the decisive part that I missed. I can also understand @MT_ Head that saying No thank you, I'm full while smiling isn't offensive at all. Thanks for that. 
I conclude that there is not a single wide spread word I can use which doesn't sound stilted or strange (or even has a touch of sexual pleasure). 
I now have a repertoire of polite responses available. I'm accepting the most helpful answer for me, but most of you got my up-vote. :)

A:

If you're looking for a single word that will express thanks at the same time, I would avoid both satiated and sated; with apologies to @F'x, "sated" is not at all common in conversation, and when it is used it usually has more to do with sex than food.  (Not always, but often enough that your hosts will look at you strangely for a moment before relaxing and saying to themselves "Oh, that's all right - it's not his first language."  Probably not the effect you were hoping for!)
Believe it or not, No thank you, I'm full is perfectly well accepted, especially if you look happy when you say it.  It may seem a little utilitarian, but it's what we USAites generally say at home.  The only time when "I'm full" would seem rude to your host/ess would be if you hadn't actually eaten much at all; the implication would be that you had eaten somewhere else first.  But I suspect that that's universal, and not restricted to the US.
If you know your hosts well, you might actually make the cook even happier by saying I'm stuffed!1 , rubbing your tummy, and smiling - but only if you are very comfortable with the family.  Again, it's very important to look happy when you say this - definitely avoid this phrase if you're feeling ill and looking a little green.
Years ago, my step-grandfather insisted that I have eaten sufficient was the only polite way to indicate satiety.  Do not do this!  If you thought "I'm full" made mealtime sound like filling up a tank, "I have eaten sufficient" makes you sound like a robot.  A really, really weird robot with no friends.

1 - Yes, I'm aware that "stuffed" has other meanings, but in this context it's perfectly clear and innocent.

A:

MT_Head's answer is spot on — saying "I'm full" isn't rude.
I don't think there is another single word that is similarly polite and well-understood.
If you want to avoid saying "I'm full", you could say things like, "I've had plenty," or "I've had too much already." 

Host: "Would you like any more?"
Guest: "Oh, no thank you, I've had
  plenty. This was all wonderful, thank
  you."

Or...

Host: "Do you care for seconds?"
Guest: (leaning back) "Oh, no I can't,
  I've had too much already. Thank you,
  it was delicious!"

Or...

Host: "Please help yourself, there's
  plenty more."
Guest: "Oh, no, don't tempt me!"
  ( in a joking / laughing manner )
  "No, I better not, thank you. This was all fantastic!"

All of these are ways to avoid saying "I'm full" which would sound natural. However, be prepared that "I'm full" is probably the keyword your host is looking for, so if they insist...

Host: "Are you sure? There's more dessert!"
Guest: "No, no I really can't. I'm perfectly satisfied right now, if I ate any more it would be too much."

In this way you can imply to your American host that you are not turning down the food because it isn't good, but because you really have eaten all you would like to eat — or even a few bites too much.
Really I think what this (sadly) reflects is that we Americans are prone to eating too much food when we really like it, so if you don't seem to have eaten "a little too much" your host may wonder if the meal wasn't to your liking.
So, while you don't have to say it yourself, it's probably best to understand that this is reason people in America say "I'm full." You're indicating to the host that the food was so irresistible you already had a little bit too much, and now you really must decline (or risk being sick!).
As an American this isn't part of my culture that I'm especially proud of, but it is what it is, and it's better to understand than to remain uncomfortable with it.
I hope this helps! Good question, by the way :)

A:

In polite company I would say something flowery like
"I couldn't eat another crumb thank you — it was delicious and I ate far too much"

