From the Google Chrome blog:

Today, we’re updating the Chrome beta channel with a couple of new capabilities, especially for web developers. Fresh from the work that we’ve been doing with the HTML Speech Incubator Group, we’ve added support for the HTML speech input API. With this API, developers can give web apps the ability to transcribe your voice to text. When a web page uses this feature, you simply click on an icon and then speak into your computer’s microphone. The recorded audio is sent to speech servers for transcription, after which the text is typed out for you.

You can try this it out yourself on Google's html5rocks.com website (you need Google Chrome 11 beta installed). 

It works on Linux - I tried it on Fedora 14 and Ubuntu 10.4 with no problems.