My 19 in. Liquid Video LCD monitor (E19LCD1) cycles repeatedly on and
off again after I turn on the computer or after I start the computer
after it has been in a Stanby mode. However, if I let the screen saver
start, and I move the mouse to refresh the normal desktop view, the
screen usually stays on fine.

My question is: what do I need to do to get this to stop? 

While I do not know much about how LCD monitors work, I am a
technician and I can probably replace the defective part if I know
what to look for.

Try turning off standby mode and see if the problem still occurs.  If
that doesn't work, then try the following:

1. Go to display properties (right-click your desktop background, and
choose properties)

2. In display properties click the Screen Saver tab

3. In the screen saver tab, click the "power" button

4. Make sure "turn off monitor", "turn off hard-disks", and "system
standby" are all marked as never

5.Click the hibernate tab and make sure the "enable hibernation" box is unchecked

I hope my advice helps you out.