I actually answered my own question, but others may be interested in what I found.
I just got a Samsung UN32EH5000, and wanted to be able to turn off the picture but leave the sound on. I discovered that if I press the Menu button on the TV's remote then use the down arrow to get to the Plug & Play options, then use the right arrow and the down arrow to get to the Eco Solution, then use the right arrow to get to Energy Saving, then use the right arrow and the down arrow to get to Picture Off, then press the Select button (in the middle of all the arrows), the picture turns off while leaving the sound on.
Just pressing the Menu button turns the picture back on.

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