The whole thing is really smooth and I think it turned out well. After I made this, I decided to try to replicate the wooden rings found in this Instructable: http://www.instructables.com/id/Rings-made-from-solid-wood/step5/Buff-them-to-completion/. However, I did not have a lathe, so instead, I used a hand drill with a 1/2" bit to make a hole in the middle of a piece of scrap wood I had. Then I took my Dremel to widen the hole to fit my finger. Once this was the right size, I then took my belt sander and Dremel and sanded the outside of the ring into a thin circle surrounding the hole. After this, I once again got out my sandpaper and sanded the ring smooth. Lastly, I put a couple of coats of some walnut stain and a final coat of Danish Oil to seal everything off. Here are some pictures of the final product:
This turned out okay, but not as refined as I would have liked. However, I think it's pretty cool overall. So, now I have a ring that I'll never wear, but at least it looks halfway decent sitting in a drawer somewhere in my house.