Do you ever get the chance to write in Japanese on a computer?
It's different from handwriting, in that with a PC you input Japanese sentences using romaji. This means you are given not only an opportunity to check your own understanding of hiragana and katakana, but also your kanji comprehension, so it can be very good practice even for advanced learners.

Do you know all the romaji keyboard rules for typing characters like ん, －, ゃ and っ? In order to make sure you do, there are some good sites where you can practise all the basics. First, try warming up using the site below.

Japanese Typing Practice for Beginners (Introduction article)

For information about installing Japanese fonts, in one of CosCom's (Introduction article) contents, Japanese Fonts, there is a section on Viewing Japanese characters properly. Please refer to this.

Let's look at some more free typing games for advanced learners.
All of these are websites aimed at native Japanese speakers.
If you think that sounds just right for you, why not give yourself a challenge using these sites. Not only can you practise your typing/test your typing ability, you can also see for yourself exactly how much you can understand of the expressions and kanji in the Japanese you encounter in the games.

タイピング無料ゲームセンター (Free Typing Game Centre)

In the introduction to typing compilation you find here, you can practise the basics of touch-typing. For the intermediate level there are lots of games, and for the advanced level you can practise inputting long pieces of writing. As this typing focuses on whether or not you can read the kanji correctly, it's also a great chance to test your kanji power!

寿司打 Sushi-DA

Sushi-DA is a game for practising typing in romaji.
Type in the letters on the screen before the sushi plate disappears off-screen, and compete to see how many plates you can collect/how much you can eat!
This is a free flash game you can play online, so feel free to visit if you have a few spare minutes in which you want to practise, or if you just feel like killing some time!

This is a typing game that uses sushi toppings, so you can enjoy looking at all the tasty dishes going past as you practise.
You might start feeling a bit hungry sooner or later, though...

In NORMAL mode (通常 Play) the game includes sound.
In QUIET mode (静かに Play) there is no sound.

FURIGANA mode (ふりがなつき Play) lets you display furigana for all the kanji - useful if there are any kanji you can't read. For learners of Japanese, this might be a good way to start off.

Typing Practice Question Screen

There are different levels of difficulty. You can choose both the number of letters you need to input and the time limit you are given.

タイピングバラエティ＜なんだろな＞ (Typing Variety: What is it?)

Number proverbs, fish proverbs, tongue twisters, Dad-jokes...
There are lots of themes here for you to choose for your typing practice. The bonus here is that you can learn some Japanese proverbs as you type!
(These contents are part of e-typing)

You can find another article in the "e! Kore" section called Is typing in Japanese a weakness of yours?! Solve your problems here! (Beginners version).
There we introduce some sites where you can do some basic typing practice.