Okay, I'm bilingual (Dutch-English, to the point where I don't know which was my first language, although my English is much more well-exercised and my Dutch is probably vaguely old-fashioned because I primarily exercise it with my elders) and I'll bite.
One of the untranslatable words in liveforever's first writeup in this node, the Danish hygge, has a Dutch equivalent, which I've always found extremely difficult (though not impossible) to translate into English: the adjective gezellig. Its noun form is gezelligheid, and it basically means "a pleasant, comfortable, happy, friendly time spent in good company." That's a mouthful, and so I prefer to say "gezellig" or "gezelligheid" if at all possible, but as liveforever points out, that doesn't make it untranslatable. However, my preference for the Dutch word is very strong: once, when spending three weeks in France with a friend, the first three days of which were spent in the company of his grandparents, who spoke only French and Arabic, I was asked what I would, in English, call the kind of happy, social, congenial companionability I was eventually able to find with these friendly senior citizens and their friends. (They had a French word for it, but I can't remember it now.) I replied that it was difficult to express in English, but that the Dutch word "gezellig" was just about perfect.
But that's enough anecdote for one writeup. The following are other Dutch words that don't translate well, which is not to say they don't; it just means I wish English had them, and so they make me prone to borrowing.
lekker This is a great word. It's an adjective that means everything from "tasty" to "sexy" to "feeling good, comfortable, relaxed, happy, and healthy." (October 23, 2001: I am happy to report that according to StrawberryFrog, "lekker" has been incorporated into English in South Africa. Hurray for loanwords and borrowing!)
leuk Similar to "lekker" in that it's a simple word for a complicated set of positives. "Leuk" means something around the intersection of "nice", "fun", and "cute" (by which I mean all the English shades of cute, from "sexually attractive" to "friendly" to "small, deformed version of something that might otherwise be termed beautiful" (see Tem42's writeup under cute for a better explication of this meaning).
Finally, I'd like to point out that if there's one class of words that's hard to translate, it's prepositions (or postpositions, depending on the language). You know what I mean. Words used to describe relative position or configuration of objects. U.C. Berkeley linguist and cognitive scientist George Lakoff says this is because all position words (or phrases, in many languages) are a combination of certain basic underlying units of meaning called semantic primes, and don't translate exactly unless words match up prime for prime (more coherent writeup to follow under semantic prime once I've gotten a chance to look at my Lakoff notes a bit). He'd probably explain words like lekker and leuk, or any of the other "untranslatables" mentioned above, as difficult to translate because they seem like fundamental units of meaning in one language when really they're a specific combination of fundamental meanings that doesn't appear in another language.