Spoon yogurt onto several layers of heavy-duty paper towels; spread to 1/2-inch thickness. Cover with additional paper towels, and let stand 5 minutes.
Scrape into the bowl of a food processor using a rubber spatula.
Add the cheeses, sour cream, and garlic, and process until smooth, scraping sides of bowl once. Spoon yogurt mixture into a medium bowl, and stir in the spinach, minced dill, and pepper. Cover and chill.