This article was originally published in December 2013, and the changes are now in effect. Please look at the update at the bottom of the article for information on how to pay by direct debit.
From 1 October 2014, tax discs will no longer be issued or be required to be displayed on vehicles. Also from that date, it will be possible to pay your road 
tax annually, every six months, or monthly by direct debit.
There is more information available here. It’s also been covered in much of today’s media. The changes do not negatively impact motorists in any way – the surcharge for paying six-monthly or monthly, for example, will actually be half of what it currently is when you pay six-monthly.
The tax disc first appeared in 1921. According to the article, over 99% of motorists pay their road tax on time.
The only question I would have is what happens if someone’s monthly direct debit is refused? Are they then untaxed? Since enforcement is by ANPR (Automatic Number Plate Readers/Recognition) systems there could be a rise in the numbers of those being stopped for having no tax – yet they might not be aware that they aren’t taxed.
NOTE: As a reader has pointed out, the term “road tax” is technically a misnomer, and the correct term is “vehicle tax” or “vehicle excise duty”. However, I should point out myself that the term “road tax” is almost universal, even to the point of being in the OED. You can read more about the debate surrounding the term on Wikipedia.
A lot of people are finding the blog on search terms associated with “how do I pay by direct debit?” The short answer is that I don’t know – not in detail, anyway. My own tax is paid automatically by my lease agent, so I don’t have to sort it out myself.
However, my understanding is that if you go to a Post Office to renew your tax, you can sign up for direct debit there any time after 5 October 2014. From 1 November 2014 you will also have that option if you renew online. You will be able to pay annually, biannually (every six months), or monthly. More information is given on the GOV.UK website here.
There is currently a beta version of the online renewal system. You can try it out here.