The photo shown below is an improvement on my previous attempt. The water dripping below the ice cube is a product called Aqua Gel that I learned about from a video Robbie posted in the first thread. It worked really great and made the task so much easier, so a big "thank you" goes to Robbie!
Setup
Essentially the same as in the other photo except that slightly repositioning the primary lamp made it possible to eliminate the reflector on the right side and the small continuous-light lamp that was shining light onto that reflector. That also made it possible to light the ice cube using a simpler setup, which then allowed more of it to appear clear. Now that I no longer needed the space previously taken up by the reflector, I was also able to draw a black curtain that prevented colors in the room from being reflected in the ice cube. The colors that are still shown in the ice cube are the result of prismatic diffraction and look natural to my eye. The Aqua Gel worked so well on its own that, unlike the first photo, I added no liquid to the surface of the ice cube.
Take 2 Original
Take 2 Revised