OK, I'll hazard two guesses here.
1) The Iray vs 3Delight thing has already been alluded to. If you are rendering in Iray, try applying the Iray Uber Base shader to the diamonds. This will keep your textures, but flip things to Iray if they aren't Iray already.
2) The max path length may be set too low. If it's -1, that's 'infinite', but any other number affects the number of times light rays bounce. I've seen mentions in a few Daz threads about setting this number less than say 8 causing black eye issues, so I imagine that translucent diamonds could have a similar issue under Iray.
Another thought, this one is a long shot. Try deleting all of the lights in a scene and using a HDRI sphere (which generates illumination from all directions). HDRIHaven.com has a few of these (just pick one and use it for the environment map). My thought here is that if the HDRI illuminates your diamond properly, then it could be a spotlight issue of some sort. Again, this one's a long shot, but might be worth experimenting with at least.
Dunno if these ideas will help, but they might be worth checking out at least.