Id be really interested in what people come up with for these too. I have numerous ones at home, approx 2 metres long each, with 20 LED's poweres by 2 x AA Batteries. Was thinking of obtaining a controller of some sort and joining multiple strings to make some sort of flashy blinky thing. Not too sure what yet. still trying source an appropriate controller.
The trouble with these strings is that the LEDs are all in parallel. Sometimes they have a current limiting resistor(s), sometimes not.
I've seen one where the batteries fed the string via a single resistor. Others (particularly the solar versions with one battery) have a little PCB that steps the voltage up a little and also provides the current limiting.
In short, you can run these from a higher voltage, but the parallel wiring means the series resistor value needs to be chosen very carefully. Also, if the PSU voltage is too high, you waste a heck of a lot of power in the series resistor.
That said, the candy canes on my mesh display element employ parallel LEDs and I run them from a 5V controller with suitable series resistors.
I have some 120 led solar lights that I've been playing with (3 ch of 40 leds each), these are all in parallel as well running off 2 batteries (3.2V).
I ran them up to 4V where they are drawing 800mA with out the need for any current limiting resistor.
I was thinking of running them off one of Rays 5V supplies as these can be wound up or down a bit. Also they will be going right at the front of my display which is the longest cable runs so I may have some voltage drop to help out as well.
Just waiting on some supplies from Ray to continue my testing.