Looking at the file you sent me, I'm trying to work out what you are using as a light source. I can see that it is above the table rather than on the camera which is a good start. I would be inclined to soften the shadows rather than eliminate them altogether as having no shadow at all looks a bit artificial. The easiest way to soften the shadows would be to bounce the light off the ceiling. Just turn the light to face up towards the ceiling, this will spread the light over a much larger area and soften the shadows considerably. Rather like the difference between a sunny day with direct light and a cloudy day.
One way to eliminate the shadows completely is to use a piece of white perspex and light it from underneath as well as from above, this requires careful balancing of the two light sources and is not for the faint hearted.
Professional jewellery photographers use a light tent. This is a pyramid made of translucent material which sit over the items, there is a hole just big enough for the camera lens. You then position your lights around the tent, this produces a soft light inside and also stops objects in the room, like you and the camera, from showing up as reflections in the shiny metal. I have made one of these out of tracing paper and garden canes which worked very well.
So to sum up the trick is to spread the light over as large an area as possible. If your walls are white this is no problem, if they have a colour you might need to correct the colour a little in the computer.