Find out exactly what the LED's need, and then pick a supply voltage high enough to give a 'reasonable' drop across a resistor - too small a drop and you don't have enough range for changes, such as caused by temperature or voltage variations - essentially the higher the resistor value, the closer to 'constant' the current flow becomes (but the greater the dissipation in the resistor).
Historically a 'constant' current source was simply a high value resistor fed from an unregulated high voltage source.
You're really better off with a proper constant current source, rather than just a resistor - something like this perhaps: