Step 3: Assemble It
I couldn't get my schematic drawing program to work, so here is a hand drawn.
The constant current source has a drop voltage of 3 V, so the supply voltage should always be 3 V higher than the LED voltage and can be up to 37V which is the maximum input voltage of the LM317.
Example: You are going to connect two white Luxeon LEDs with 3,42 forward voltage each (mostly mentioned as Vf in common datasheets). The input voltage can change from 9,84V (3,42 + 3,42 + 3) till 37V (3,42 + 3,42 + 30,6).
You can connect up to ten high power LEDs to this circuit.
The higher voltage you supply the LM317 with, the hotter it gets. so it wont be a good idea to supply it with unnecessary high voltage.