Farmer Dan shared this melon cutting technique with us yesterday:
as shown: Cut the melon in half. Place it down flat. Cut slices down from the top following a "rib". When the melon is cut this way, all the seeds are clustered at the top of slice, and you can just scrape them off (or spit all of them out in your first bite).
These melons have tasty flesh right to the edge of their thin skin. By cutting narrow slices (1 or 2 ribs wide), you can easily scape out all the meat with your teeth.