I have seen these circular attachments for sewing machines http://www.youtube.com/watch?v=180NDoeRGDU This one is the Brother one but Bernina and Janome both make them as well.
My DH is going to have a go at making one that clamps to my sewing table instead of the sewing machine. You would get a perfect circle every time.

Cut the circle bigger. Take a running stitch around the edge. Put your heat proof template in place and draw the thread so it gathers around the disk. Spray with water and iron. Take out the disk. Pull out the thread. Pin in place and sew around the folded edge.

I use a feather weight fusible inter facing (Joanns). Cut circles same size as circle templates. Put glue side and right side of fabric together, sew around the edges. clip a slit on the inter facing side, turn right side out. Then I can place on my back ground fabric and iron into place. Then you can stitch into place and have no raw edges. This is how I did the circles on the centers of my Dresdan Plates also. You get a nice crisp rounded edge to your circles.

I have used freezer paper...cut the finished size you want, cut fabric 1/4" larger, you can press the extra fabric around the freezer paper with your iron. I then use invisible or matching thread to use a sort of blind stitch or very narrow zig-zag to attach to background fabric. I then cut out fabric behind the circle, and the freezer paper can be removed at that time. Makes very smooth edged circles!

When I make circles I use coffee filters! Sew it attached to the right side of the fabric. Use pinking shears if you have them to cut around the outside edge-snip the middle of the coffee filter and turn Perfect circles every time!!

finished edges
 

Was at 1/2 priced books yesterday. Saw a quilt book with lots of patterns using circles. The edges were covered with bias tape. No fraying; certainly would be able to match your fabrics in your block by making your own tape. Can't remember the name of the book . Sorry.

Measure to the right of the needle and tape a thumb tack with point up. Put the tack about 1/2" larger than 1/2 the diameter of the disk. Place the fabric with center punched down over the tack. Then slowly stitch with basting stitch turning the fabric as you stitch. This will give you an even circle of stitching. When finished then gather up the fabric around the disk. Tie thread and even out the edge. Spray edge with starch and remove the disk. You should have a good circle to then applique to base fabric.