Break open the chiles and remove the seeds and veins. Put the chiles to cook in a medium sized pot. Cover with fresh water and gently boil until chiles are very soft. Let the mixture cool and using a favorite method, blend the chile and the water to make a paste and strain.
Meanwhile, put the cubed pork, oregano, garlic, onion and salt into a large heavy pot and cover with water. Boil meat gently for 30 minutes. When the meat is soft, add the chile and hominy and cook for 15 to 20 minutes until the mixture is boiling nicely.
To serve, ladle the posole into heavy bowls and serve with thinly sliced cabbage and radishes, quartered limes, oregano, chopped onion, and fresh corn tortillas. Besides these side dishes, posole is usually served with sodas or cervesas.