Here's the short version: multiply your source vector by a 3d rotation matrix. Zaxo's snippet is nifty, but not quite sufficient as it doesn't do matrix math.
The extra complexity that comes with moving from 2d rotations to 3d rotations is the fact that, in 3d, you don't really have a canonical axis to rotate around. In the xy plane, when you do a rotation by r radians, you pretty much have to rotate around the z axis. In 3d space, you can rotate around the x, y, or z axis, or around an arbitrary vector (if that gives you problems, think of it as rotating around several canonical axes at once). So you can't easily describe a 3d rotation with a single parameter; you basically need a matrix (or a quaternion, but let's not go there).
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