These are quick and simple rings made form small scraps of hardwood turned on a lathe.
The rings pictured here are Walnut and Hickory. I haven't tried these out of soft wood such as pine, but I do not think they would work to well so stick to a hardwood.
These really are very simple, the hardest part is getting the sizes right. Once you have the process down you can crank them out quick.

Step 1: Gather Some Scrap Wood.

Go through your scrap box of exotic woods, find some hardwood with nice tight grain. I used pieces 1-1/2" X 1-1/2" Squares about 1/2" thick and 3/4" thick hickory and walnut.
You will also need a larger junk piece of wood attached to your lathe and turned with a taper to a dull point that will hold the ring by friction (jam-chuck). I am using some scrap pallet wood. A 2X2 would work fine for this. I actually think a softwood would work better for this.

Step 2: Drill the Hole

Now we need to drill a hole the proper size for the finger you want to put it on. A 3/4" hole barely goes on my pinky, and simply falls off my Fiance's largest finger. Just experiment, these are fun to make.
http://en.wikipedia.org/wiki/Ring_size

You can drill the hole a few different ways. I used the lathe with a jacobs chuck in the tail-stock. and used my chuck to hold the piece while I drilled.
Its always best to drill a pilot hole if you are drilling a large hole. The pilot hole will guide the larger bit through the piece. So I started with a 1/4" bit and followed with a 3/4" bit.

You can just as easily use the drill press for this.

Step 3: Prepare a Ring Chuck

I do not know what you would call this jig other than a ring chuck. It is simply a short spindle of wood turned round and then to a dull point at one end.
*** UPDATE*** It's called a Jam-Chuck

The point to this is to slide the ring on to the spindle snug enough so it grips the ring tight enough that you can turn it round and shape it.
If you push the ring blank on too tight, you risk splitting the wood. If you go too loose, you risk the piece coming off So go a little on the looser side. here is a small tip: If you make the spindle long enough, the ring blank will simply come loose and spin sloppily. simply turn off your lathe and re-snug it to the spindle and try again.
Also note, that if you go too tight, it may not break until you make the ring thin.
snug it on and turn the spindle to correct rough size then rough sand. Check the fit of the ring blank often and stop when the ring blank goes on nice and snug.

Step 4: Turn the Ring to Desired Shape

Time to shape the ring to finished size and then sand...
I used my Carbide lathe finisher from start to finish. This has to be done at a high speed and very, very light cuts. Go slow, you really don't have to much to remove. High speed, slow and light cuts!
A nice Idea here, is to make a match set of rings using one solid piece of wood. After you get it finish smoothed, simply part it any way you want to. The grain will match each ring giving a somewhat more sentimental value or "friendship rings".

Step 5: Buff Them to Completion

After sanding all the rings thru 400 grit, I buffed them with three stages. Then finished them with some wax . The oils from your fingers will keep them protected.
They are very very cool.
The pictures sadly do not do them justice at all.