The art and science of Texas Barbecue. Meat Selection. For the most part, cuts of meat to barbecue vary widely from region to region. Traditionally, beef and pork cuts are most common, but poultry (chicken and turkey) is widely prepared, and lamb and goat are regional favorites. Barbecue has its roots in using tougher, less valuable cuts of meats although today some barbecue establishments even cook prime rib as one of the menu items.