I hope this will help you set-up Japanese input, it’s really easy.
I think it is a similar process for other Ubuntu-like linux distributions. Well, here it is:

1. Click on the Linux Mint Start menu icon.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_001.jpg

2. Click on “Control Center” then open “Language Support”.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_002.jpg

3. If it asks you to install a Language Pack, click Install, and wait for it to finish.

4. Under the “Language” tab, click on the “Install / Remove Languages button ”

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_004.jpg

5. From the list find Japanese (Press J to instantly find Japanese) and click the check-box next to it, then click “Apply Changes”. Close it after it’s done.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_005.jpg

6. Now go to “Language Support” again (by following step 2), and change the “Keyboard Input Method System” to “ibus”.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_006.jpg

7. A little Language Input icon should appear on your task-bar now. Click on the little language input icon and click on “Preferences”, here under “Input Method” tab.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_007a.jpg

8. Click on Select Input method > Japanese > Anthy.

http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_007b.jpg

9. Now click “Add”.
http://i240.photobucket.com/albums/ff82/ultraali453/Textfugu%20-%20Post/jap_007c.jpg

10. Now you should be able to type in Japanese!!! Just press Ctrl + spacebar to switch between English and Japanese. ^_^