I decided to, after two years or so, once again check out what has come of Runescape - but, predictably, that didn't go as smoothly as I had hoped. Even after installing Sun Java and the proprietary ATI fglrx drivers (both are a requirement for reliable working of Runescape and many other Java and non-Java games, by the way), it simply would not go into OpenGL or Software mode, and constantly fall back to Safe Mode with the message "Runescape was unable to enter that display mode". After some searching I figured out that the issue was in the Java browser plugin, which apparently failed to use OpenGL properly. Had I been using Windows or OS X, it would've been easy as a stand-alone client is offered on the Runescape website. However, there is no such thing for Linux - at least, officially there isn't.
Someone has apparently written his own 'port' (wrapper, actually), to run the stand-alone client on Linux. It supports resizable mode, HD, and OpenGL mode - so basically everything you need to play Runescape HD. To get it running, just do the following and follow the instructions:
wget http://dl.dropbox.com/u/11631899/opensource/Perl/unix-runescape-client.tar.gz tar -vxzf unix-runescape-client.tar.gz cd runescape/ ./runescape
It will automatically update with the latest version from the Runescape website, and launch the applet stand-alone. Et voila, problem solved :)
(additionally, a similar approach probably works for other OpenGL Java applications as well.)
Update (September 6, 2012)
HikariKnight, the author of the wrapper, has responded in a comment below, clarifying about the issue and giving an additional tip for users of his wrapper. I have added emphasis and edited some things for readability.
You however forgot to mention that you can add application menu entries by running ~/runescape/install-desktop-icons - that way you don't need the terminal to be open. You can also update it by running ~/runescape/update-runescape-client .
About the OpenGL issue: The reason it kicks into safe mode, is because Java 7 (as well as OpenJDK 7) have been compiled without linking the plugin libraries - meaning it has no idea where to find the native Java libraries. This results in the application either crashing, or being kicked into safe mode.