Following the instruction from http://www.pablo-bloggt.de/linux/twitter-programm-fur-ubuntu-tweetdeck/, I go directly to http://www.tweetdeck.com/desktop/.

But when I press the 'Download Now'-Button, it's Label changes to 'Tweetdeck wird installiert' but nothing else happens.

Edit:

I found this which explains that I have to look for alternatives to Adobe Air.