I don't mean to sound mean, but why is it DIYers always want to wire this way??
WHY not just run the feed to the switch??? WHY complicate things?

If you really want to wire it this way I think you are describing it correctly. I am not totally clean on your terminology.

The black from both cables will get spliced in the light box.
The whites will get spliced to the white from the light also.
The reds will get spliced to the black from the light.
AT the last light you can use 12/2 if you want, but the power must be brought down on the white and back up on the black, which would then get spliced to the red.
To keep it simpler for you you certainly can use the 12/3, just keep the same wiring pattern as you had been.

One more question; WHY 12/3?? You will be MUCH happier using 14/3 if this is a new circuit or if you are feeding from an existing 15A circuit.
Remember, bigger is NOT always better.

Is this 12/3 with ground OR 12/3 including ground.

The OP does not mention a red wire at all and his mention of wiring is in series. You are going to have a lot of dim bulbs if any light at all.

I see that he says "12 gauge 3-wire" and not "12/3 wire". You may have a point Hendricus.

Cables do NOT count the ground in the number of wires.

12/3 would be black, red, white, AND GROUND (often bare).

12/2 would have a black, a white, AND A GROUND.

Cords count all the conductors.

NO, NO, NO!!!!! You said: "My understanding is the black will be continuous to the switch and the white will be broken by each light fixture." That is a SERIES CIRCUIT. In house wiring, all things EXCEPT switches are wired in PARALLEL. Switches are wired in SERIES. If you hook up your lights the way you just described, they will be very dim, as each one will only get a small portion of the voltage.

You should put the switch at the start of the run, not the end. You will use 14/2 or 12/2 cable depending on the size of the circuit breaker. 12 is for 20 amps, and 14 is for 15 amps. Run the cable from the breaker box to the switch. Then run a calbe to the first light. At the switch, connect the two wite wires together. Put the switch between the two black wires. Now go back to the first light. Connect the black and white wires from the cable to the black and white wires on the light. The light will now work. This is the basic circuit.

For the next light, run a cable from the first light to the next. In the box with the first light, connect all the blacks together and all the whites together. Go to the box for the second light. Connect it just as you did the first. Black to black, white to white. Repeat this for the whole string.

In the image above, I left out the grounding wire. All grounds must be connected together in each and every box. If the boxed are metal, they must also be connected to the ground wire.