I have taken some pictures of rings when I was doing up my website a year ago.
www.proposalring.com
(Which reminds me... I have to start updating the site again.)
I used a compact camera then, and this was the result:
http://www.proposalring.com/ringGallery1.htm
Unfortunately, I did not have knowledge of things like the depth of field, white balance, etc. some 2 years ago. So, the photos were very far from perfect. But as you can see, you need light to bounce off the diamond to show off its beauty. A diamond is a multi-faceted light reflector. Without light, it just looks dull.
If you want to see 'fire', which is multi-coloured light, you need to shine tungsten light into it, preferably from many angles.
If you want to see a nice white diamond, then you need to shine white light into it. For that, you need studio lighting or a macro ring light. But a cheap and very acceptable solution is sunlight.