Yes, it's possible to access, buy, and download music from iTMS on Linux.
If you're not averse to spending a little money for a fantastically useful app and to running a proprietary, nonfree program (but then again, if you were, why are you using the DRM-laden iTunes Music Store?), CrossOver Office from CodeWeavers is a great way to get your iTMS fix. As a bonus, you'll also be able to run other Windows-based, proprietary programs, such as Microsoft Word, Excel, PowerPoint, and Internet Explorer, as well as multimedia apps like QuickTime and Shockwave. The advantage of CrossOver is that it makes it really easy to run these Windows programs in Linux, and you won't be skirting the terms of their software licenses. The disadvantage is that CrossOver costs $40, which is, IMHO, a pittance for all that it offers you. If you're not sure if you want to buy the program, you can try it for free for 30 days, which should be plenty of time for you to kick the tires and discover how useful this program really is.
To start with CrossOver, head over to http://www.codeweavers.com. CrossOver Office is featured prominently on the home page, with a big red Try Now button that you should press. Follow the instructions on the page and download the software via FTP, BitTorrent, or HTTP.
After downloading the CrossOver shell script, you need to make it executable and then install it. Both actions are performed as a normal user, without sudo:
chmod 744 install-crossover-standard-demo-5.0.1.sh ...