mozc is an open-source fork of Google IME and the convertion precision is very good. The only short point is when you login to the machine, it starts with DirectInput mode. You need to change it to Hiragana mode. But gnome-session-flashback, the lightest but full-featured desktop environment, does not have that screen.
So if you use gnome-session-flashback, you need to stick with ibus-anthy or ibus-kkc (the "Kana Kanji" in the figure). kkc is developed by one of the developers of ibus-skk, so it is still good.
If you want to use ibus-mozc, you can use mate, a fork of GNOME2.