One specifically useful Twitter application available for Linux is Choqok, a more than adequate open source social networking tool that offers the added functionality of StatusNet and Open Collaboration Services integration.
With all of the usual 140 character status updates, retweets, direct messages, etc. that you would expect to find in a Twitter app, there are few desktop Twitter apps across all major platforms that are more accomplished than Choqok, which can be installed via your usual sources or in Ubuntu via the Software Center.
Your preference might be for something a little more comprehensive however, which will require something with a little bit extra. That extra is Adobe Air, and the tool concerned should be TweetDeck, a Twitter client that will display multiple accounts (as well as accounts from Facebook, MySpace, LinkedIn, Foursquare, Google Buzz) making it very easy for you to stay on top of your various networks and statuses.
To use TweetDeck, begin by installing Adobe Air (Ubuntu users can find this in the Software Center) and then visit www.tweetdeck.com to download and install!