If you're tripe isn't fully pre-cooked, place it in a large pot with the onion, carrot, celery, salt and enough water to cover. Simmer for a good hour or more, until almost tender. (Or use a pressure cooker, using the "stew" function or set at high pressure for 20-30 minutes.)
Drain the tripe and run fresh water over it. Leave it to cool and drain in a colander. When the tripe has cooled off, pat it dry with paper towel and cut it into bite-sized strips. Set aside until needed.
Now, in a terracotta or cast iron pot, gently sauté the onion in abundant olive oil until soft and translucent. Add the garlic, bay leaf and oregano and give it a stir, let everything sauté gently for just a minute, then add the tripe and give it a stir, too, so it's well-covered with the olive oil and aromatics, seasoning with salt as you go. Stir in the wine and let it evaporate, then sprinkle in the sweet pepper powder and stir, so that the tripe is entirely coated with it.
Add the tomatoes to the pot and enough water (or, if you prefer, broth) to barely cover the tripe. Cover and simmer until the tripe is fully tender.
When the tripe is nearly tender, add the potatoes along with the peperoncino. Top up with water if needed and continue simmering until the potatoes, too, are tender. If using, mix in the pieces of peperoni cruschi to the pot a few minutes before the end of cooking.
Let the dish rest off heat for 10-15 minutes, then serve.