I forgot to put this in my other post, but I think that the traditional discrete solution is the Voltage-reference (or LDO) into Current reference circuit. Basically you can turn any LDO or plain old voltage regulator into a current reference by setting their feedback in a different manner.
That doesn't work for me here because I do have a 1V minimum voltage requirement. But most people probably are working closer to 3V or 5V.
You can mirror a current generated by a LDO over somewhere else though. The current mirror methodology is truly quite flexible.