You can't remove the reflections, but you can change what gets reflected. This can be achieved by moving and shading things and lights around you.

First of all, as long as you're shooting shiny things with on-camera flash, you'll always get ugly reflection right in the middle of your picture. You can remove it only by polarizing the flash and/or lens, or by moving you main light somewhere else.

The first step is usually getting a tilt/swivel flash and bounce it off the ceiling or nearby wall. Then it's matter of finding out which one of those reflections is not visible in the object or at least looks best.

If what you're getting by this point still looks ugly, you might have to get more lights and move then around, or use the last resort -- light tent which makes all of the surroundings the same invisible white.

But not all reflections are bad -- you can use them to show the nature of your object (e.g. show that it's reflective). So keeping one or two strips of light can be beneficial.

Unfortunately I don't have any good examples of reflections of light at hand, so I'm going to post examples of dark reflections.

This is my first attempt to photograph a glass thermometer thingy. You can see that surrounding it with white doesn't work very well.

What works is putting two black boxes to the sides. They are reflected on the edges of the glass, and make it possible to actually recognize what we're looking at.

Therefore I think you should not be avoiding the reflections, but rather use them to help your picture. Unless it's that big image of on-camera flash in the middle, or course.

---

update: Lighting diagram for the first shot. I think this setup can be thought of as poor man's light tent.