How do I turn off the border around Unity launcher icons?

The launcher icon overlays such as borders and highlights can be removed quite easily by changing 4 icon overlays in Unity.

1. Make a backup of the following image files in the folder /usr/share/unity/4/:

   launcher_icon_back_54.png
   launcher_icon_edge_54.png
   launcher_icon_glow_62.png
   launcher_icon_shine_54.png
   

2. Either make your own transparent images of the same size or download the files at

   wget -O transparent54.png "https://i.stack.imgur.com/MiFpK.png"
   wget -O transparent62.png "https://i.stack.imgur.com/0NvP0.png"
   

3. Replace those 4 files with the same 4 files downloaded from the above link.

4. Restart unity by pressing Alt+F2 and typing setsid unity (or logging out and in again).

5. If you don't like it then just replace the icons with the backed up icons and restart unity.

Example:

It is possible, and rather simple.

1. Install compizconfig-settings-manager:

   sudo apt-get install compizconfig-settings-manager
   

2. Then run about:config in the run dialogue in Unity: press Alt+F2 and enter about:config.

3. CCSM will pop up with the Unity plug-in open. Navigate to the experimental tab:

   

4. Now change the Backlight Mode to "Backlight Always Off".

5. Problem solved.

In light of your comment, the box never 100% goes away (by design).

Small note from @ccpizza concerning 16.04.1.

• In ubuntu xenial 16.04.1 the Experimental tab is gone, and the setting is now under the Launcher tab and the dropdown is called Launcher Icon Backlight Mode.

I scripted the removal of the boxes around the applications and the one around the Ubuntu button (for 12.04):

cd /usr/share/unity/5
wget -O transparent54.png "https://i.stack.imgur.com/MiFpK.png"
wget -O transparent62.png "https://i.stack.imgur.com/0NvP0.png"

for name in squircle_shine squircle_edge squircle_base squircle_base_selected launcher_icon_back launcher_icon_edge launcher_icon_shine
do 
    sudo mv ${name}_54.png ${name}_54.ori.png \
        && sudo ln -s transparent54.png ${name}_54.png
done

for name in squircle_shadow launcher_icon_glow launcher_icon_shadow
do 
    sudo mv ${name}_62.png ${name}_62.ori.png \
        && sudo ln -s transparent62.png ${name}_62.png
done

Then to restart Unity, press Alt+F2 and run setsid unity. The result looks much cleaner to me.

Before:

After:

To switch back:

cd /usr/share/unity/5
for name in squircle_shine squircle_edge squircle_base squircle_base_selected launcher_icon_back launcher_icon_edge launcher_icon_shine
do 
    sudo rm ${name}_54.png \
        && sudo mv ${name}_54.ori.png ${name}_54.png
done

for name in squircle_shadow launcher_icon_glow launcher_icon_shadow
do 
    sudo rm ${name}_62.png \
        && sudo mv ${name}_62.ori.png ${name}_62.png
done