As of this month, the 2×16 character LCDs from Jee Labs are 100% suitable for 3.3V use. The previous batch was made for 5V, although the backlight can always be used at a lower voltage: it’s just a few LEDs with a current-limiting resistor after all.
But the logic level officially had to be 5V. Then again, in practice, I’ve used those LCDs with a LiPo battery without any noticeable problems.
The new LCD units are specifically for 3.3V, so both voltage jumpers on the LCD Plug should be set to +3V.
To see which unit you’re using, look on the back. These are the new 3.3V units:
Note the “33V33” suffix.
I suspect that the difference is only in a few on-board resistors. I haven’t tried running these new LCDs at 5V logic levels – they may not be up to it.
But for LiPo and 3x AA battery use, the new units will be perfect.