Lets just spill the solution right away:
JTextArea text = new JTextArea();
There is also a built-in solution, which is that Ctrl-[Shift-]Tab still works. Actually, those keys always work. Those keys are also the solution when you suddenly find yourself focus-stuck within a JTable. I first thought that a JTable constituted a a "focus cycle root", but it doesn't - it is just the same effect as with the JTextArea problem here outlined (and can be "fixed" the same way).
However, Ctrl-Tag does not quite give the feeling one typically is after, and in particular not when one use the JTextArea more as a solution to text that typically is less than a line big, but can become multiline. Such a field thus looks like a JTextField, but the Tab character suddenly have a completely different meaning.
I Googled a lot, and there are lots of solutions (or rather hacks), but didn't find the above solution, which seems to be the most straightforward after Java 1.5. The huge heap of questions and solutions should, IMHO, give a hint to the Swing team that a specific method for a JTextArea to not suck up Tab events should be made available.
nullmeans that it will take the focus keys from its parent.