Turns out I had to change an obscure registry setting:
Open regedit and go to:
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Video\
In there, there'll be some keys looking like this:
{ECA904C2-25E6-4680-9B2C-44EBC0EC9190} (with different numbers and letters ;o)
Highlight the "Video" key at the top of the tree, and do a search for "1920x1080" (or whatever resolution you want to change).
You'll find some keys in subfolders named 0000, 0001, etc.
The keys will begin with DALR6 DFPXXXXxYYYYx0xFF
XXXX= Horizontal resolution
YYYY= Vertical resolution
FF= Display frequency
Before you continue, take a backup of the original keys by right-clicking the {xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx}-type key, and select "Export"
Click on the key matching the screen resolution and frequency you want to remove the borders of.
A lot of the key will consist of "00", but there'll be a few non-00 entries. Change all of these to 00.
Finally, I had two subfolders, 0000 and 0001, containing mirrors of the same keys. When I changed a key in one subfolder, it also changed in the other.
This fixed the problem for me, no need to install CCC.
Effects will take place next time you change the screen resolution to a resolution you changed.
Note that doing this is at your own risk!
Actually, just changing the first non-00 entry in a key will remove the issue, but when mirroring my desktop @ 1920x1080x60, it'd reinstate the borders. Changing all the non-00 values to 00 fixed this. I have no idea what the entries do exactly, all I know is that it worked.
/Fangrim