1) For this design, first open a couple dozen 5mm (I.D.) brass jump rings and close a few dozen 3.5mm (I.D.) silver-plated jump rings, just to get rolling. (It's not a bad idea to start your chain mail projects this way; it'll help stave off the frustration of having to steadily pick up and put down your tools.) I like using two different colors for this design so it's easy to keep track of where you need to connect.
2) Slide six of the closed silver-plated rings onto one open brass ring.
3) Use two pairs of chain-nose pliers to close the brass ring. Set the assembly down.
4) Pick up three silver-plated closed rings with one brass open ring, but don't close the brass ring yet.
5) Connect the open brass ring to final three closed silver-plated rings you added to the last ring.
6) Repeat Steps 4 and 5 to the desired length.
7) It's a pretty nice chain now, but I decided I wanted it to be a little stiffer, so I added one more set of brass rings so that the brass rings are doubled up.
Here's the final result.