Mosfet constant current
Here is an alternative method that is almost as simple and is more repeatable (and, probably at least as important, has a lot of compliance): 







simulate this circuit &ndash; Schematic created using CircuitLab

Use a MOSFET rated to work on 2.5V drive if your logic input is 3.3V. It needs to turn on fully with about 2.7V to get the full compliance range. You can also use a BJT in place of M1. 

The current is about 0.6V/R2. 

It works by Q1 reducing the gate voltage to M1 (or base current if a BJT) as the voltage across R2 approaches 0.6V or so and Q1 starts to turn on. 

It will drop out around 0.6-0.7V or so if the parts are well chosen, so a 3V LED can be maintained at fully brightness/current to about 3.7V input.