I'm not a huge fan of some of the things Adobe has released over the last couple years, e.g. the uber-bloated Acrobat Reader, but I do like their Air product. Sadly, until a couple weeks ago it was unavailable for linux... but that has now changed!
Adobe Labs have released an alpha of Adobe Air for linux, and tonight I finally remembered about it and downloaded it. It is closed source, but I find it useful, so I installed it. Installation is as easy as 1) download the bin file, 2) give it execute perms, and 3) run it as root so it can install systemwide.
I then proceeded to the Twhirl website, and while that site's easy download button did not yet realize that Air was available for linux, there is a direct download link, which Firefox opened properly with Air and installed (it asked for my password again to install as root).
From there, simply click on the icon, enter your Twitter username and password, and away you go!