None of the linux buildbots run with X enabled. Consequently none of the tkinter (or tkinter user) gui tests are run on Linux. It was thus pointed out to me, during discussion of using ttk widgets in Idle, that we do not really know if ttk works on the variety of Linux systems (beyond the one Serhiy uses) and that I should look into this.
I asked on python-list for help, by linux users running python3 -m test -ugui test_tk test_ttk_guionly test_idle
Seven people did so with Debian Jessie, Debian Wheezy, Gentoo, Mint, openSUSE, and Ubuntu (x2). One machine failed once with the ttk test, and then passed. Another failed the tk test until a mis-configuration was fixed. So tkinter, and ttk in particular, seems to be working on linux.
Do any of the core devs who run the test suite on Linux do so with -uall or -ugui? The gui tests above take about 10 seconds (mostly the tk test) on my machine, with some flashing boxes near the end.