I prefer to use the Adobe AIR applications and I use/used the following on Linux (Ubuntu 8.10 and OEL5):
- Twihrl: Single column, colorful, multi-account supporting, simple application, good for small / low resolution screens
- TweetDeck: That is my current stable preference (despite it is beta). It is a whole deck of different columns providing replies, home your user groups. Also stock twits, comprehensive search and compact usage. Can update Facebook too. Some minor problems exist. Unfortunately it does not support more than one Twitter account. You cannot rename your groups (which I would prefer Seesmic Desktop for this reason)
- Seesmic Desktop: That is still at the alpha level and many features might not be available (like favoring items, low cost search). Like tweetdeck, you can have groups but also you can rename and manage them easily. There are a lot of bugs (like you cannot clear or refresh groups) and unfinished functionality. One good feature is that it can handle many Twitter accounts at once.