This appears to have a valid concern (increased risk of scratching the screen when laid flat), but has been mistakenly extended into a blanket recommendation with no basis (nebulous concerns about TVs/monitors "going bad").
How did I come to this conclusion? I figured one way to go about this would be to look into LCD TV/Monitor manufacturer's manuals to see if there were any warnings about this. While not guaranteed, I'd figure those who make these devices should know about this and at the very least have a vested interest in not being liable for returns if customers broke products and no warnings were given.
Well, pictures say it much better than words:
- LG LCD TV manual (LINK) :

- LG W2753VC LCD Monitor manual (LINK):



(No mention of how to transport in terms of orientation, just that using original packing material is recommended.)
- Panasonic TC-L42U30 (LINK):

- Sharp 70" 70LE732U (LINK):

Based on all of these images, it seems that we can draw a couple of conclusions:
- The pure orientation component cannot have anything to do with this "folk" recommendation, otherwise manufacturers would specify that laying them flat would be disastrous.
- Several manufacturers (all except one) recommend putting a soft cloth down under the screen. I see two possible reasons for this:
- Put something "cushy" under the screen so that if there are any irregularities on the surface, it doesn't damage the screen, and because a cloth won't scratch the screen
- Support the screen/glass to prevent deflection which might damage it. There are discussions on many forums about the glass potentially breaking if laid flat (HERE for an example).
I can't tell from the manuals whether cracking is the concern, or whether it's scratches/uneven pressure.
To see if cracked glass was a real concern, I tried finding complaints online about this issue. After all, if this truly happens, individuals should be complaining and asking for input after it happens to them! Alas, no such luck. While absence of evidence isn't evidence of absence, I would have expected to find more online in forums if this was a common occurrence.
Takeaway: In the end there's no mention whatsoever of concerns about the horizontal orientation of monitors except for scratching the screen. Nothing about sagging potential, cracked glass, electronics, having liquid crystals go bad, etc. If laying a monitor flat was that big of a concern, I'd at least expect a mention in the caution sections (I looked at them in all manuals above) of even a possible issue. Nothing of the sort. Also, there's no bread crumb trail of real users complaining about their stupidity for breaking their precious screens. Therefore, I see no basis for the concern.