I picked up a pair of inexpensive tile nippers from Home Depot, per YHL's recommendation.
And then I attempted my first tile. I made sure the nippers were squarely centered on the tile and started applying slow pressure, as John recommended. And then I gave it a good push...
...and successfully cracked my tile. Oops.
I tried again, but this time I put tape across half. Still no dice.
Then I flipped the tile over, and decided to try snipping it upside down, using the existing grooves along the back as my guide.
Booyah!!! I ended up not even needing the tape once I got the hang of it.
One thing I highly recommend is covering your tile and nippers with a paper towel or hand towel, as you are snipping it. This will prevent teeny shards of tile from flying everywhere.
As I started dry fitting my penny halves into the fireplace, I realized I would need a little more variation in size. So, I started cutting several intentionally uneven.
I filled in the outside edges, and am SO happy with how it turned out. Next steps are to figure out what I want to do with the grout (remember my big grout dilemma awhile ago?), and finish off the inside edge. It's amazing to think this project is so close to completion!