Are there any good, free 3D game engines similar to UDK that are under open-source licenses without copyleft provisions?
Panda3D is relatively nice (note: relative to most FOSS game engines, not things like UDK) and free/unencumbered, though you would need a commercial license for FMOD if you use that I think (FFTW and ARToolkit also need commercial licences if you use them, but those are more special case tools). FMOD is dirt cheap as middleware goes though. Unity isn't open really, but I don't think there are any royalty requirements on it.
Cube 2 is an engine similar to quake engines that is under zlib (soon CubeCreate will be released with lua scripting based on Cube2)
There is also the more general Panda3d, Ardor3D, jMonkeyEngine. As well as a handful of 2d engines you can use without a problem
However there is nothing close yet to Unity, UDK or Source yet