The secret is...
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(Drumroll)
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You go in to Filters-->Other-->Offset
Enter in the width and height boxes exactly HALF of your graphic's width and height.
Toward the bottom of the mini-screen that pops up, make sure that "Wrap around" is selected.
This ensures that your graphic will connect with itself on all 4 sides! All you need to do then is make the distinct lines INSIDE your image (should look like a + sign) go away -- use the smudge tool, connect any squiggly lines, etc.
Once your graphic "looks good" again, you're done! NOTE: I would recommend against doing any editing/modifications on the edges, or you'll be "back to where you started".
To show how easy it is, I made a few myself!
Matthew