Line 2 heavy large baking sheets with aluminum foil.
Whisk the oil, oregano, and pepper in a small saucepan over medium heat just until warm.
Set aside for 15 minutes. Brush the oil mixture over both sides of the tortillas. Stack the tortillas and cut them into 1/2 to 3/4-inch wide strips. Arrange the strips in a single layer over the baking sheets. Sprinkle the cheese and salt over the strips. Bake until the strips are golden brown and crisp, about 12 minutes.
Do-Ahead Tip: The tortilla strips can be made up to 2 days ahead. Store in airtight containers and keep at room temperature.