Oh yes, you still can. The LM317 is able to take anything from 0-40V as an input. Becareful that you choose the right resistor values so you don't have the LM317 taking up alot of current. That way it would not heat up.
The purpose of the circuit is to regulate the current using the transistor (I'm assuming you are asking about the circuit that I drafted up). And the transistor is controlled with voltage from the LM317. So as long as the pot is of suitable value, and adjusted correctly, you can still have a low current flowing through the transistor, and you can run a red LD with no problem. The only caution is what I've said: be careful not to let the LM317 overheat by calculating the behaviour of the circuit first.
Why I'm using 3 standard 1.5V cells or 1.2V which I hope to do so finally is more of an individual preference. I have GP AAA batteries at home with a decent charger, and am not willing to buy a CR123 charger, and rechargable batteries just for this laser, as I have no other household appliance using CR123.
I basically want a laser powered on AA or AAA batteries, and I looked around DX. I couldn't find any nice looking 3x AA torchlights, but I found 3x AAA torchlights, hence this build.
It's also good for exploring low dropout regulator circuits, because although I used an LM317 that has a high dropout voltage, the overall circuit's effective dropout voltage is none. I have turned it into more of a current circuit with a max current limit inherent due to the parts. I'm currently trying to improve that.