This weave is also known as the parallel weave. I would suggest using 6mm and 8mm rings to begin with. This weave can also be done with 5mm and 7mm rings.

This sample is done with 6mm, red and 8mm, green/purple rings. Not the prettiest in color combos, but some rings I had laying around.

Step 1. Start by placing 4 8mm rings onto 2 6mm rings.

Step 2. Add an 8mm ring, pink, in between the 8mm rings of step 1 and around the 6mm rings.

Step 3. Now you will add 2 more 6mm rings, blue, and 2 more 8mm rings, silver. Then add your 8mm in between and around the 6mm rings. That's all there is to this one. The hardest part is adding the center, pink, 8mm ring.