Unplugging home TV saves power?
Is the sentence "Unplugging appliances that aren't being used saves
  power." still valid for modern devices?


Yes, this is not limited to "older" devices. To make it easier on yourself, plug them into a power strip and use that to disconnect all of your devices.

The amount of standby power drawn is small but finite. This will depend on the device, however. The multimeter and oscilloscope next to me draw zero power when switched off (hard disconnect from the power lines), however the monitors in front of me draw power no matter what, even when switched off since they have a 'soft' power switch.

Take a look at this link to see a standby power experiment http://standby.lbl.gov/summary-table.html