1. Tunnelbear. Even the name is cute. Tunnelbear is a VPN service that lets you use the internet as if you lived in a different country. The Little Bear version is free, but to watch the Downton Abbey episodes you will probably have to spring for the $4.99 a month version. That may or may not be worth it to you, and there are other options that may be free. After you install it, set the dials to On and UK. Make sure it is connected, it may take a bit. Then go to http://www.itv.com/downtonabbey/ and watch the latest episode after it airs in the UK. Episodes may only be available for a short time, so be sure to keep up with watching it. Thank you to graspingforobjectivity.com for this tip!
2. Sidereel. Last fall I used Sidereel.com to watch Season 2 before it aired in the US. I would go to the http://www.sidereel.com/downton_abbey page and click the "Search" button. Other links would come up that I could then use to watch the episodes. This is somewhat unreliable and subject to the links being taken down.
3. Delishows. Another web site with links to Downton Abbey shows is Delishows. Click here to see what episodes are available after they air in the UK.
4. Google. If the above fail trying searching on Google for other options.
Edited 9/7/2012 to add:
5. Expat Shield. I have read of people using Expat Shield with success. Download the free software, and enable it. It then gives your computer a UK IP address, so you could watch the episodes on ITV directly after they are uploaded, which is usually within 24 hours of airing. Make sure to disable it after you are done watching or your will be automatically be redirected to UK sites when you open amazon.com or other web sites.
Edited 9/10/2012 to add:
6. FastpassTV: Cat (see comments section below) told us about FastPassTV. This looks like another helpful site to try!
I'm glad everyone is getting so excited for Season 3! I know I am!
Disclaimer: This is meant as information. It is up to you to judge for yourself if there are legal and moral issues to watching these episodes online before they air in your own country. This information is theoretical and may or may not work in your situation.