There are currently a few ways to obtain Relic Pieces;
1) Find Relic Pieces in the Abandoned Ruins.The Abandoned Ruins are the main areas for mining in My Time at Portia, and you can find both ores, Data Discs, and Relic Pieces here. All the different ruins will give you Relic Pieces, so make sure to go there even if its one of the ruins you don't generally prefer. The best method for getting Relic Pieces is to use the Relic Detector, since this item will allow you to press F to see where the Relic Pieces are located.
The Relic Detector can be upgraded to be even more useful, and this 5,000 Gol investment is great for hunting Relic Pieces.
2) Find Relic Pieces in Lucky Sacks.The travelling NPC called The Mysterious Man sells an mystery item called a Lucky Sack (as well as other items of course). Each Lucky Sack cost 1,000 Gols, and typically cotains a single random Relic Piece. However, the sack can also contain other mining-related items, so you might not find a Relic Piece every single time.
Buying Lucky Sacks from The Mysterious Man is a great way to get Relic Pieces if you got a lot of spare Gols laying around, and prefer to spend that instead of mining.
If you don't know how and where to find The Myserious Man, make sure to check out this guide!