Last week, I read CHIRI Yukie (知里幸恵)'s collection of Ainu kamuy-yukar or "songs about gods", 『アイヌ神謡集』 (Ainu Shinyou Shuu), and man was it an interesting book.
Background: according to this page, Ainu Shinyou Shuu is the "first genuine record of Ainu chants by an Ainu". Printed in (roman character) Ainu with a Japanese translation on the facing page, it was compiled by Chiri shortly before she died at the age of 19, and published posthumously. Chiri was, in (my translation of) her own words, "born an Ainu and grew up in the Ainu language" ("アイヌに生まれアイヌ語の中に生い立った"), and according to this timeline she had been working in the field of Ainu linguistics and mythography since the age of 17, alongside giants of Ainu studies like KINDAICHI Kyousuke (金田一京助) and John BATCHELOR.
There are some words in Chiri's Ainu text that are obviously loaned from Japanese, like shirokani for silver. The Japanese word for silver is shirogane (銀 or 白金), made up of shiro (white) + kane (metal, esp. gold). This is obviously well beyond coincidence, and it ain't likely that the Ainu version was the original (unless you want to argue that Japanese borrowed the word, broke it into two parts, and then made each one a fundamental lexemes). Similarly, "gold" is konkani, and I would bet you a thousand yen it comes from Japanese kogane (黄金).
Other places where the two texts coincide are sake, as in the alcoholic drink, and pashui for "chopsticks" (hashi in Japanese). Both of these words are found all the way back to the Kojiki, the oldest surviving Japanese text*, so their exact etymology isn't easy to figure out**, but since traditional Ainu culture didn't include rice farming and chopsticks were invented in China (meaning that the Japanese speakers got them first), it seems safe to conclude that the Ainu were the borrowers in this case.
In a collection like this, though, the king of intriguing vocabulary has to be kamui, Ainu for "god" (or "totem spirit", "nature spirit", etc.), which seems suspiciously close to Japanese kami, "god". But kami is another Kojiki word, and refers to a concept all cultures have, so the question of who borrowed from whom is more difficult to answer.
According to my Nihongogen Daijiten, it has generally been argued that kami as in god (神) is etymologically linked to synonyms referring to other things that are high: 上 as in "upper" or "above" or "superior", 髪, "(scalpal) hair". If you believe this, you probably have to call kami the predecessor of kamui, if only because it's more general.
On the other hand, the ND also notes that the pronunciations of 神 and 上 are consistently distinguished in the old texts, courtesy of the 8-vowel system of Old Japanese. To be specific, the i in 上 is an A vowel, but 神's is a B vowel. (In fact, this B-vowelness is thought to be reflected in the extra u in the Ainu version.) It also points out that Japanese kami are not necessarily located "above" humans. "We have to call this one 'not yet settled', but the argument for separate origins seems stronger to us," is their conclusion.
To summarise: kami's roots remain uncertain and it follows that its relationship, if any, with kamui is also murky. But, as far as I can tell from looking through various dictionaries and webpages, the generally accepted hypothesis is that the Ainu word is borrowed from Japanese. I'm not sure why everyone assumes this -- maybe just because the Japanese-speaking community has been more powerful than the Ainu community for more than a thousand years (and -- to put it mildly -- has tended to abuse that power, unfortunately... but that's another story).
So are there any beyond-a-doubt Ainu words used by Japanese speakers? Sure, but the vast majority of them are place names in Hokkaido, the northern island to which the Ainu were gradually driven back over the years and therefore the last part of Japan to be settled by Japanese speakers. The name of the capital of Hokkaido, Sapporo, probably comes from an Ainu phrase referring to a big river (either dried up or running through a plain -- exact details vary from source to source).
Most of the other Ainu words in use in Japanese tend to be names for animals and plants. (Speaking of which, bonus trivia: the fashion magazine non-no got its name from an Ainu word for "flower".)
As for the actual content of the myths -- well, I'm even less qualified to talk about Ainu mythology than I am to talk about Ainu linguistics, but, if you read Japanese, you can enjoy the whole book online (it's waaaay out of copyright). There are also parts of it online in Esperanto, of all languages. But if you prefer your reading material in English, check out the Ainu section at sacred-texts.com. I especially recommend MIURA Kiyoko's collection, which overlaps a lot with Chiri's book. (And I advise you to take Arthur Waley's [alleged] translation of Kutune Shirka, "the Ainu Epic", with an extremely large grain of salt.)
A lot of the stories revolved around the exploits of a guy called Okikirmui, who (unlike his older cousins) is "smart as a god" and indeed often delivers savage beatdowns to gods who mess with him -- beatdowns which end in the gods being assigned their animal form. "I was unpleasant to Okikirmui (or his sister), he kicked my ass, I passed out, and when I woke up I was stuck between the ears of a dead frog/otter/etc." is a very common complaint. Good times.
* Of course, some later texts, e.g. the Manyoushuu, do contain a lot of material originally composed before the Kojiki.
** Although I personally am completely convinced by the Nihongogen Daijiten's tentative linking of hashi to the verb hasamu ("to capture between two objects").