First let me caution that I am no Ubuntu Guru!

I have Karmic Kubuntu 64 on one of my systems and once had Tweetdeck on it.
I can't recall why I took it off, probably just wanted a simpler solution,
but thought I'd try re-installing to see if I could duplicate your problem.

So, like you I'm sure, I went to the Tweetdeck website downloads section and
clicked on the download button. I had cut right to the chase by choosing the
"download" item from my Google search, not completely sure if I was going to
wind up with a Windows executable, but all worked just as it should. In
short, it installed Adobe Air and the application which worked the first
time.

The only tiny glitch was that there was a window requesting permission to
use my saved password which was hidden under other windows.

So I can't help you, except to suggest that you remove everything and try
again. It does work for me on my 64-bit Karmic Kubuntu machine. I can't see
how it could be any different on Ubuntu, but having said that I know
different hardware/software configurations sometimes do conflict.

I'd happily remove Adobe Air (TweetDeck doesn't appear to have managed
to install anywhere that I could find), but can't seem to figure out
how to go about it ... any ideas anyone?

I managed to get the Air uninstalled - took a couple of reboots
before I managed to find it in Synaptic ...

Went to the [TD] website and told to install ... asked if I wanted to
install Air first ... yes please ... all seemed to install ok,
launched, then threw up a dialogue saying that this was "one of a
small machines" Air is unable to run on, and they're working on it (or
something like that - I didn't make an exact note of the wording - too
busy swearing at it ...).

More reboots & uninstalls later, back to square one ...

sounds like good old microsoft windows IT strategies...

Now that you've thrown down the gauntlet so to speak, I suppose I'll
have to set up tweetdeck on ubuntu for drill. I know linux doesn't
require reboots for software installation - though a logout/login might
be needed for some changes to take effect.

I'm guessing that, for some reason, things didn't get installed
fully/properly (probably related to the "air vs my machine" thing),
and the restarts have somehow managed to get it to register into
synaptic properly (registers as "adobeair" (all one string ... or, at
least, it does here) ... and it's not like "air" is something that's
easy to misspell, so I can't have missed it that way ... to be honest,
the reboots with the final uninstall were done simply because I'd had
to do them the first time round, not from not finding the software in
Synaptic ... I worked from "experience", and rebooted to save myself
any further stress ...first time round, though, I definitely *did*
have to restart a couple of times before it showed ... bizarre ...
however, first time round, I *did* install air from a magazine
coverdisk ...

Well now you know! I'm in the same position trying to use 9.10 on my MSI
Wind. Blinking display brightness is very annoying. Roll on 10.04!

I had absolutely no problems with my aspireOne ... mind you, I *did*
originally install 9.04 (using the UNR distro, which gives you the
option of the funky UNR gui, or a standard gnome front end ... I stuck
with the gnome) first, then ran the upgrade when 9.10 appeared - no
hitches, though ... try that route ...