PPSSPP is the most popular way to play MHP3rd today. The save data is stored in a user-specific directory:

The save data system for Monster Hunter Portable 3rd is a case study in aggressive Digital Rights Management (DRM) that ultimately punished legitimate consumers who wished to backup their progress.

Recommendations for Users:

Monster Hunter Portable 3rd represents a unique entry in the Monster Hunter franchise regarding save data management. Unlike its predecessors and successors, MHP3rd features a save system that is strictly tethered to the specific hardware unit it was created on. This report outlines the file structure, the mechanisms of copy protection, the impact of "save hacking," and methods for data preservation and migration.