Recomended method, from the repositories
You don't need for anything TeamSpeak from the download pages. The package is included in Ubuntu multiverse repository. Just make sure that you have the repository activated and type in a terminal:
sudo apt-get install teamspeak-client
You might need to clean previous installations of the client. If you run into more problems you might need to report a bug.
Own your own risk, 3rd party repository
From the TeamSpeak forums, twprogrammers offered a method to install TeamSpeak 3 using his repositories.
NOTICE: Under no circumstance you should add and install software from anybody you don't trust. If you are aware of the risks please continue reading. I'm in no way responsible of the possible damage to your system.
The instructions offered by twprogrammers couldn't be more clear:
Adding the repository to the sources:
sudo sh -c "echo 'deb http://apt.twprogrammers.com linux net' >> /etc/apt/sources.list.d/twprogrammers.list"
With this we have created our list file for apt.
Adding the GPG key to your trusted keyring:
sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 1BCCE0BF
In the official site they use
keyserver.twprogrammers.com as keyserver, but it could be down any time, so I will be using Ubuntu's.
Update your package list and install TeamSpeack 3:
sudo apt-get update
sudo apt-get install ts3-client
Notice that the name of the package is
ts3-client. That's all.