I built my own driver out of an LM317, a couple of them in fact. There are 2 ways to rig an LM317, either in voltage regulation, or current regulation configuration. I run mine from a single lithium cell with no issues, at least not for average power diodes like red. One of my projects is an adjustable diode driver I can use to test diodes, and to figure out how much current they require by adjusting my test driver until the diode reaches the lasing threshold, then I measure that on my dummy load. Works well enough. Here is a link to a set of calculators for both configurations.
http://diyaudioprojects.com/Technical/Current-Regulator/
They show the way to wire them up as well. I used it in current regulation configuration. I don't recall how much the voltage drops, but not enough that it won't drive red diodes from a single cell lithium battery.
Note: I just noticed that they have split these into 2 different pages, so I fixed the link above to the current regulator page. Here is a link to the voltage regulator configuration page:
http://diyaudioprojects.com/Technical/Voltage-Regulator/
But you want to run it in current regulation configuration. Also, for R1, I used a 25 ohm linear pot (a bit much, but it's what I had around) with a one ohm resistor in series, that way even when the pot is turned to zero, there is still one ohm, keeping the max current at 1.25A.
Since I am doing it, I got out the test driver to measure the output. My 18650 is kinda dead, showing 3.55V. But with that, my LM317 test driver puts out 3.0v no matter where I set the pot. I'll do it again and add the figures when the battery is charged. But the drop in current reg config seems to be about .5v.