The J is pronounced like the French J in jour, bijou, jeunesse. The first syllable is unstressed and therefore the first o sounds pretty much like a w in English (i.e., a semi-vowel). The stressed syllable is the final ão which is a nasal diphthong. You could try pronouncing the ow in cow with a nasal sound. It's tricky to get it if you can't hear it - actually it's tricky even if you can hear it. Try saying Jwaong quickly remembering what I said about the J sound in Portuguese and with the stress on the nasal a.
As Macunaíma put it, this is a very tricky one indeed, and I would go as far as saying that non-native speakers should be proud of themselves if they manage to pronounce "João" exactly like a native one. I have nothing to add to what Macunaíma has said, save for a slight remark on the fact that the ão syllable is a diphthong. It is a diphthong all right, but the three vowels uttered together (o+ã+o) may make them sound like a triphthong most of the time.
In this forum's Resources sticky, there are links to some websites where you can type a word, and listen to how it's pronounced in Portuguese. Look them up here.
If you are familiar with phonetic notation, "João" is pronounced [ʒwɐ̃u̯] (the [ɐ̃] should have a tilde on it, which stands for nasalization). As the other posters have noted, the "j" is pronounced the French way. The "ã" is a nasal vowel similar to the way you pronounce the interjection "Huh?" in English. Each "o" is short, with a sound similar to the vowel in the English word "do".
Thanks, everybody. I went to one of the "voice demo" websites, and found the most amazing thing: I've been saying it right! I know some French and a little Spanish, and between the two, I got lucky!
Sample text in Brazilian Portuguese Todos os seres humanos nascem livres e iguais em dignidade e direitos. São dotados de razão e consciência e devem agir em relação uns aos outros com espírito de fraternidade. Listen to a recording of this text
Since I live in Montreal, Canada, and my name is João, some people have shown interest in knowing how to pronounce it. I tell them "relax, say 'juin' (June, in French), then place your mouth as if to pronounce 'ou'(F), or 'w'(E), without actually pronouncing it, or hardly.