Looks like every redraw anywhere on screen causes Compiz to clear and redraw the whole screen, so if I close everything the flickering will be triggered each second, because of the clock which updates every second
Tried disabling all additional plugins (got plugins-extra and -unsupported package installed) - no effect.
Tried changing workaround options, turning all of them on and off - no effect.
Purged xorg edgers PPA (although I am on Intel GMA HD4500 and it never made problems) - no effect.
Even removing Unity and starting just with Compiz does not make the flicker disappear, so I am quite sure the problem has something to do with the most recent Compiz update on Precise 12.04.
The previous Compiz version was working without any flickering.
Unity-2d works without any problems.