Question: I'd like to render a very simple 3D scene in a java applet. I could do all the math myself and render that to a bitmap, but as I'm sure I'm not the first person in history to have to draw a few cubes to the screen, I was wondering how this is usually done.

Every place I've read has said either Java3D or LWJGL, but as I understand it these rely on platform-dependent code and need to be installed separately from Java, making them both unsuitable for graphics.

So how is this usually done?

Answer: 
In a plug-in 2 JRE (1.6.0\_10+), you can use JWS extensions for embedded applets. This is important because JWS makes it much easier to ensure the correct binaries are on the run-time class path. AFAIU JOGL and Java 3D both offer JWS extensions for their APIs.

Should you choose to go that way, please make the applet has the draggable parameter added, so the end user might choose if they want a (mostly useless) browser window wrapped around a 3D animation.
I'm not a Java guy, but 2/3 years ago when I was at univeristy I had the same problem :)
I've tried jogl... and for noob in Java like myself I found it too complicated to configure and write something using it.

I used Java3d and it worked like a charm, the only problem I had back then, was that Java3d did not worked with all graphic cards. But fortunatelly it worked with computers at my univeristy :)