From a Russian blog Slovomania I learned that in the new versions of X11 there is a typography layout option, similar to Ilya Birman layout. This is its Linux counterpart:
To enable it in GNOME: System → Preferences → Keyboard. Then Layouts tab, Options.... Enable Key to choose 3rd level (right Alt will do) and Enable extra typographic characters under Miscellaneous compatibility options.
Also in Russian: Типографская раскладка в Линуксе.
