If you are a Twitterholic and need a nice desktop client for Linux, Polly may be one of your best choices. Though there are great many Twitter client available in Linux like Hotot, Gwibber etc, what sets this apart from others is its simplicity, great many features and extremely lightweight memory usage and fast response. A screenshot below.

The client supports multiple streams where you can have your Twitter stream, your Tweets, mentions, Tweets and profile of someone and even worldwide trends. You can also expand and view activity (retweets, favorites etc) of each tweet. The client also supports expanded view of Tweets.

Though the app is still in pre-alpha, I haven't faced any major issues with it yet. You can install Polly by issuing the following commands in Ubuntu:

sudo apt-add-repository ppa:conscioususer/polly-unstable
sudo apt-get update
sudo apt-get install polly