Same issue here. I am using Ubuntu 12.04.1 64 bit and I can not see flash videos too.

Did either of you try the solution mentioned in #36?

Visit this link it has an explanation how to do it.
https://support.steampowered.com/kb_article.php?ref=1493-GHZB-7612
If you are not comfortable with the terminal you can do it this way.
Go to the adobe site and download a version of flash for a 32 bit linux distro in a .tar.gz format, make sure to download this to your folder named whatever you system is named for easy access during the next step.
You will need Sudo access so hit Alt F2 a window should pop up asking for a command type gksudo nautilus hit enter
you will get a prompt for you system password enter it then a window will open up this window will give you sudo rights to do what you need to do.
Go to your filesystem on the left side of the window click on your home folder then click on the folder in there titled whatever your system is called. Copy the 32bit version on flash. Go up to the top of the window click on View, in the menu that drops down you should see Show hidden folders click that. Now you will see a whole bunch of folders with a . before them look for .steam. Once in . steam go to bin in this bin folder create a folder named plugins.
Paste flash and extract it into this folder delete the original un-extracted file leaving the other three, now close steam and restart you should have flash in your steam now.
The flash that you download for 64 bit ubuntu is 64 bit flash but steam is a 32 bit program and cannot run the 64bit version you are using on your ubuntu.

For the original poster I havn't ran KDE before apparently the GUI sudo command is
kdesu
I'd assume the other steps are the same if not similar.

Workaround script: http://steamcommunity.com/app/221410/discussions/0/864958451629025044/