Japanese video reports about the Olympics often show the screen label 北京五輪 'Beijing Olympics'. I had assumed that the Japanese news media were simply following Chinese usage, with 五輪 gorin ('5 rings/wheels') being a clever and concise sound-symbol translation of the word Olympic, which is otherwise rendered more lengthily in Japanese katakana オリンピック. A Google search on 五輪 returns 18,600,000 possible results, while a search on オリンピック returns 27,000,000 results.
Well, it turns out that this usage of 五輪 for 'Olympics' is restricted to Japanese. The official Chinese homepage for the Beijing Olympics uses four characters strictly for their sound values, 奥林匹克 (ao-lin-pi-ke), a string that Google finds on 17,500,000 pages. Korean usage also relies on a phonetic transcription, 올림픽 (ol-lim-pik), with 23,300,000 Google results.
So it looks as if the Japanese may have coined the term 五輪 for 'Olympics', probably on the occasion of its 1964 Olympics. That headword shows up in my father's battered old New Pocket Japanese-English Dictionary (Kenkyusha), which was revised in 1964 after first appearing in 1958. (Does anyone have an earlier edition?) It's a clever coinage, with several positive allusions, but first I'd like to note that the Beijing Olympics logo also incorporates a bit of clever usage of Chinese characters. The stylized image of a runner strongly evokes the character 京 jing ('capital') of Beijing ('north capital'), and the logos for the various event types also evoke the old Chinese seal scripts widely used in decorative engraving from as early as the Han dynasty.
I emailed Matt of No-sword about this, and he suggested that one positive association of 五輪 is Miyamoto Musashi's Book of Five Rings (五輪書), which involves competitive strategy and tactics and is sometimes translated as The Art of War. Indeed, many of the Chinese-language search results for 五輪 referred to that book. Other Chinese results referred to "5-wheel" (off-road 4WD) vehicles, the 5th round of 6-party talks, and the popular Japanese singer Itsuwa ('5-rings') Mayumi (also known as Wulun Zhengong in Chinese). Just about the only Chinese-language usage I could find of 五輪 for Olympics was by a Chinese blogger in Japan.
A more spiritual association of 五輪 occurs in 五輪塔 (gorintō), the 5-element pagoda. And 五輪 is also homophonous with 五倫 (Jp. gorin, Ch. wǔlún), the essential 5 relationships of Confucianism.
POSTSCRIPT: I've been watching far more of the Olympics than I had planned, and I must say that I am most impressed by the good sportsmanship of the athletes from both the host country and the largest guest country. It's a big improvement over the behavior of both such parties during the 1988 Olympics in Seoul. (And congratulations to Constantina Tomescu-Dita! Her gutsy move paid off.)