While trying to find out myself, I saw Mark Phippard's comment about the General page for decorations. I went to Preferences > General > Appearance > Label Decorations
and began disabling decorations until the green bar went away.
Turns out it was Decorate Classes with Test Case
that caused the green bar, which can be unchecked to disable it. Searching for that, I found MoreUnit to be responsible: http://moreunit.sourceforge.net/
Looks like "MoreUnit decorates files which have a test case". I'm guessing it does so by seeing if there is a class starting with the same name and ending in "Test" or whatever the MoreUnit config might include.