Are there specific types of plastic that are more "Clear" than others?
You're probably looking for acrylic.

Computer case manufacturers often use acrylic. For a recent example, the Thermaltake core P5 atx case uses acrylic. Also, according to this datasheet acrylic has higher optical clarity than polycarbonate and can be restored to full clarity by polishing.

Also, acrylic sheets can be sourced easily, can be cut with a scorer, and are light and strong.