Boot with the 12.04 Live CD and see if you get sound there, if not
then there's possibly a problem with the sound hardware but worth
trying a 10.04 Live CD to verify whether it's specific to 12.04.

Did you used to have sound on it ?

Might be work rolling back to a previous kernel too in case it's a
Kernel update that's screwed up. Hit [ESC] on boot to get the menu to
select alternate kernels.

Different desktop environments work with different sound front and
backends. Some work better or worse with different hardware. You
may have to experiment a bit. I would start by looking at hardware drivers
(additional drivers). It may want to update or say you have proprietary
drivers available. If that fails then I would try a different backend. Many
distributions use PulseAudio on top of ALSA. You can try removing PA and
using only ALSA. Some people find that if the remove PA it clears things
up. Others find that if they use Xine over Gstreamer it fixes problems.

You can try installing Kubuntu-desktop from the repositories and switch at
login to see if sound works there. The reason to try this is because KDE
(the Kubuntu desktop environment) uses a different sound system. It uses
Phonon and Knotify. If the sound works in KDE then you know it is
likely something at a higher level. If it does not work then it is likely a
driver or low level issue.

It is a confusing situation. See this chart:
tuxradar.com/.../how-it-works-linux-audio-explained

There were no additional drivers found for sound. I'll give KDE a try. How
do you remove PA or use Xine?

I only have a 9.10 CD, but there is sound with that Live CD. I hit ESC and
tried to load Kernel 26.31.14 and got the message "Cannot execute - no such
file or directory".

I think I'm going to wait for my new HD, scrub Ubuntu, load Windows and
then reload Ubuntu. Maybe I should order a 12.04 CD in the meantime.

9.10 is very old now but since it gives sound this points more towards
the sound system itself not being compatible with the later versions
of Ubuntu. Some system details here would be a good idea, i.e. make
and model number as well as how much memory.

Have you tried this?
http://ubuntuforums.org/showthread.php?t=1979191
Here is a video too.
www.youtube.com/watch\
jb8vyUGE9-k&v=S6M2GjvI8EQ