First have a stylus. It's a tool that helps to pull and hold fabric. In the case with the circles it helps pull the bottom fabric to meet the top fabric and avoid puckers. I used the instructor's wood stylus and just loved it. Quilt in a Day did have a metal stylus, but I ended up with a Clover awl instead. The awl is very similar except that the tip is sharper then the wood stylus. It does have a great handle just like the one on their seam ripper. I'm on the hunt for a wooden stylus to include in my "tool box".
The second thing is to put the flat piece on the top of the piece that has to be eased in the most. I'd always done this the opposite way so I could see the potential puckers, but as it turns out you have a tendency to want to manipulate the fabric too much. You still need to pin (use five pins) starting from the center then both ends and then between these pins. As long as you keep with the 1/4" seam those pesky puckers won't intrude into the stitching. The stylus and the needle down also provide some extra holders. I even got to the point of assembly line sewing. Yep felt much more comfortable with the process at the end of the day.
I've got the borders to put on yet. This is the class sample that I plan to use as well. I've got enough of all the fabrics so I'm thinking I will use the three browns. My project is very Zen like so I'm thinking it will be gifted to Tammy since she's got her bedroom decorated Zen. Don't spill the beans.