My first attempt at a block is not bad. It could use some work, but I've definitely found the technique I'm going to use for the rest of the quilt.
When I first started thinking about making an Orange Peel block, I considered the different ways the components could fit together. I decided that I definitely wanted each almond shape individually pieced into a square, so that the primary and secondary block patterns would be identical.
l tried piecing a couple of blocks with patterns I made. They came together fine, but I didn't like the extra bulk at the corners where the solid pieces overlapped. So, I decided to try piecing the almond shapes into a solid square, similar to Dale Fleming's Six Minute Circles, and loved the results.
Note: Dale Fleming's original technique is described in the linked tutorial and in more detail in her book, Pieced Curves So Simple. I'm going to show you some photos of how I adapted the technique to my purpose, but please refer to Dale's original tutorial and book for more details on the basic pinless piecing technique (which is amazing!).
I gathered the materials I would need which, for the almond shape I was making, included a compass and a 5.5" square ruler.
Instead of making the freezer paper template described in the linked tutorial, I traced my 5.5" square ruler onto freezer paper and then drew in the quarter-inch seam allowance. (So, basically, I drew a 5" square inside a 5.5" square.)
I then used my compass to draw the almond shape. To duplicate this, set your compass to draw circles with a 5" radius (or, more simply, so that the point sits on one corner of the 5" square with the marking end just reaching the other).
Note: Freezer paper comes on a roll and doesn't easily lay flat. That's why it's taped to my table.
The finished almond shape looked like this.
l then cut out the almond shape, trimmed the freezer paper template to about 6" and ironed it to the wrong side of a piece of Kona Snow.

For the almond shapes, I cut 5.5" squares, as they should be easy to match up with the 5.5" square I drew on the freezer paper template.
When cutting the notches, I made a cut at each point, as shown above, and then pressed the notches outward toward the two sides of the almond shape.
I started sewing at one of the points and sewed around one side of the almond shape first, backtacking a couple of stitches at the beginning and end of my seam.
The tutorial says to use a zipper foot, but I just used my (very narrow) regular piecing foot.
Here's what it looked like when one seam was finished. I then repeated the same process on the other side.
Once both side seams were sewn, I folded back the solid fabric and made a few stitches (forward and backward) to reinforce the points.
I trimmed away excess fabric in much the same way I would have for a circle, trimming away the tips of the points at either end of the almond shape (much as you would if clipping corners before turning a pillow or bag right-side-out).
I ironed toward the center, starting with one side. I tried both clipping and not clipping the print fabric and found that I preferred the results when I did not clip the edges of the print fabric before pressing.
I then pressed the other side toward the center. As mentioned in the tutorial, pressing the seam allowance toward the center gives the appearance of an applique.
I trimmed my block components down to size by using my 5.5" square ruler. (Note the placement of the diagonal line on the ruler.)
Once I had four block components, I was ready to sew them together! My original sketch had the block components coming together in circles like this, but I decided to piece this blocks in a starburst pattern instead, to test how much bulk was coming together at that center point. (Answer: Not too much!)
Here they are sewn together! This was actually the most difficult part of the process, since the bulk of the curved seams fought my sewing a scant quarter inch seam allowance right up against them.
You can see here that they aren't quite aligned perfectly, but I'm hoping my technique will improve as I work on more blocks!