Jewel Pieces are items you get from Collecting. You can only find them by Collecting them. You need 10 of one type of Jewel Piece and take it to a Collecting Manager to transform them into one Jewel. The Jewel Pieces will disappear after you combine them into a Jewel.
How to Collect Edit
Go to a Collecting Area, and buy a Collector from a Collecting Manager. Then going into the Collecting Area with your Collector equipped. Then click Start on the box that pops up when you have a Collector equipped. You will then start Collecting.