They do that in New Zealand. Your employer(s) send all your income details to the tax department and at tax time the tax office just sends you a bill. Pretty convenient by the sound of it.
Japan does that, even further. Your employer not only files for you but also calculates all adjustments and pay you (if you paid too much tax) or deduct from your wage (if you owe some). If you work for one company and have no extra income, which had been typical for majority of Japanese until recently, you'll never have to deal with the department of taxation.
But I think it has downside. Those who do not touch their own tax calculation don't tend to think about taxes; most of my acquaintances when I was in Japan had very vague idea of how much they paid for tax. After moving to US I noticed that people are much more aware of tax they're paying, hence how the government spends them.
I now even think Japan should make everyone file, like US.
Norway is very similar. They send you a return that is already filled out, and if you agree with the numbers, you don't have to do anything. If you do need to make changes (such as, say, the 10% credit for foreigners during their first two years in the country) you go online and add the information yourself.
The Netherlands isn't much different. You can download a program (even a Linux version :)), download the data they have on you, load it, add info and submit the form. All I did was check that the income was correct and I filled in the amount of interest I paid for my mortgage. Result: they owe me 15 EUR.
