Even though personally I am not a fan of Adobe Air applications and prefer native Twitter clients on Linux, I cannot ignore an amazing application such as TweetDeck which comes with some very interesting features like real-time Twitter streaming - a feature that has been added for all users (it was previously enabled in a test version) in the latest TweetDeck 0.36.
This new feature will update most of your columns in real-time so you may want to turn off the notifications if you plan on using it. Why "most columns"? Because Twitter may throttle your search results under certain circumstances. Also, the Twitter Lists are not yet included in the real-time Streaming API.
While this feature is enabled by default, you can turn it off from the TweetDeck prferences (under Settings > Twitter - uncheck the "Enable realtime Twitter streaming" checkbox.).
The latest TweetDeck also supports the new Twitter "Snowflake" tweets - a change in the Twitter API which will break all Twitter clients which do not support it starting with November 24.
Download TweetDeck 0.36 | Adobe Air installation instructions for Ubuntu (see only the first part of the post).