What is an effective way to clean clear LEGO bricks? What kind of cleaning products are good and safe to remove smudges and finger prints off clear LEGO parts? Normally I would just use the Clorox wipes. The parts in question are 1x2x2 panels in clear plastic.
Almost all LEGO can be washed with dish soap and warm water (Do not use dish washer soap though, it sometimes contains grit to help grind off dried food. This will scratch your bricks)

Second best option is windex, which isn't too bad for plastics.

I would advise against clorox wipes though. I believe they contain bleach, which while is great for getting things clean, it will react with the ABS plastic. 

I tried bleaching some old lego a number of years back, and I soaked the bricks for a day or two and I found that, while they wer very clean, they also slightly expanded as a result of reacting with the bleach. They are very sticky, and are hard to separate now. Additionally, they seem a little harder and less flexible.

Note that this occurred after being immersed in a water bleach mixture for extended time. One swipe of bleach will probably not have much effect, but it shows that it does react with the plastic, so that is why I suggest against it.

Warm water and dish soap cures most problems.