(disclaimer: I work at Microsoft but not on OneNote.)
Great, I thought and went right over to get it for my Kubuntu box. Oh right, not that sort of cross-platform.
But they offer a web version that tells me "Take your notes anywhere with OneNote" (except to a Linux desktop of course!). They're clearly big on this "anywhere" idea, something about accessing your notes from anywhere ... and then "The add-on works only in 32-bit Internet Explorer, Chrome, Firefox, or Safari (Mac).".
It's no different to other companies but these things do start to grate after a while - "access from most browsers" or "access from most OS" obviously isn't as good a marketing tag as "anywhere".
/rant
It seems I can use it from my 64bit browser on Kubuntu but then it operates like Evernote's web app does.
[This is just how it is from my angle, really just feeding back in case others on Linux would have looked it up. Dare-say you can get it to work with WINE?]