It runs Silverlight through Wine as a plugin in Firefox/Chrome on Linux. Works flawlessly for me even with HD video.
As for the future, I believe Chrome already supports the underlying technology, and Firefox confirmed they will be adding it in the future, so they will probably support Netflix's HTML5 DRM at some point. Although I'm not sure that it's any more of promising for Linux compatibility than existing propritary software - it will still need to be compiled specifically for each platform, and may even have some OS level requirements too (not sure on that, but the fact this is only working with Windows 8.1 and OS X 10.10 makes me think so). So Linux may still be left out and not be able to run certain DRM extensions.
Before someone says "Chrome OS is not Linux": well it is Linux-based, it runs the Linux kernel, and the Linux version of Chrome. I am not aware of any of our (Google's) internal plans, but I presume the intent is to eventually make all the HTML5 Premium Video Extensions functional on plain Chrome with no dependencies on bits of software currently in Chrome OS. This would make streaming possible on all Linux platforms, providing Netflix doesn't artificially restricts playback to only Chrome OS. (I am also not aware of any of Netflix's future plans).
"Should" or "would"? Not likely, because it will be easily circumvented. There are also more fundamental reasons why linux and DRM are at odds: https://www.fsf.org/news/fsf-condemns-partnership-between-mo...