Did you try removing the steam redundant lib32 files? (yes this is a terminal command)

find ~/.steam/root/ \( -name "libgcc_s.so*" -o -name "libstdc++.so*" -o -name "libxcb.so*" \) -print -delete

You may need to run this again after each major update as steam loves downloading those old libraries (it also priorities them over newer ones).

PS. AMD 290x cards are not Linux friendly IMO, I have a 390x and performance is facepalming bad for allot of games, give it a shot anyway and let me know your experiences!

Thanks for the help... I'll try and tell you the results. And yes, you're right... the 290X is performing very bad on Linux... it's a shame till it's a very potent card. I hope that the coming new drivers by AMD will be more sofisticated. It will sadly take some time... aprox. 6 month, I read. And I even don't know if they will support the R9 200 cards :-(

Sorry I didn't reply earlier. I've been busy in the weekend.... So to the problem itself:

It worked! And I really have to delete the libraries after every Steam update. And Steam will always tell me, that the installed version is outdated and that I should get the newest one. Is it to Valve to fix this problem or is it Ubuntu related, only. What kind of libraries are these?

An alternative to deleting the bundled files is to use LD_PRELOAD to specify that the executable should look elsewhere for them. I've modified the .desktop file for steam to use this command.
sh -c "LD_PRELOAD='/usr/$LIB/libstdc++.so.6' DISPLAY=:0 steam"
This method will continue working after steam updates itself.