This method uses iBus which is the default and easiest input method to install. It does not require complicated setup or terminal commands.
First, make sure language support is correctly installed. From the desktop, click on System / Administration / Language Support.
If it tells you that the language support is not installed completely, click install.
Click on ‘Install / Remove Languages’. Select Japanese and then tick ‘Input Methods’ and ‘Extra Fonts’. Click on close.
Change ‘Keyboard input method system to ibus’. Click on close.
The correct files should now have been installed automatically. All you need to do now is configure the the input method. First, log out and then log back in again.
To enable the Japanese input system you have just installed, select ‘System / Preferences / Keyboard Input Methods’. Drop down, Japanese, Anthy, Add.
There should be a keyboard icon in the top right of your screen. You can click on it to change to Anthy (Japanese input) or use Ctrl+Space. Open a text editing program, change to Anthy and you can now type in 日本語。
If you want extra options, click on the keyboard (ibus) icon on the top right and select preferences. Then change the ‘show language panel’ to ‘When active’. This will pop up a further language bar that you can use to tweak Anthy.