Not common on most tourist menus, but the quinto quarto ('fifth quarter') of the animal, whether veal, lamb, beef, goat poultry or whatever, has always had a big role in tradtional cooking. If you look you can find it in simple peasant stews and also in celebrated recipes such as the Piedmontese finanziera or vincisgrassi from the Marches.
Offal is made into a score of regional sausages, such as the Puglian turcinieddri. Trippa (tripe) is most popular in Tuscany, as are budelli (chitterlings) and obscure organs such as lampredotto, the fourth stomach of a cow.
In Naples and elsewhere in the south, frattaglie are an old-fashioned street food, now seen mostly at festivals, sold from little vans: o' pere e o' musso, 'trotter and snout' and other organs besides, boiled and chopped up in a paper cup with a slice of lemon.