Rich meats need acidity and tannin to help combat the fat and intensity of flavor they bring to the table. In fact, rich meats are ideally suited for young red wines if they are particularly fatty, like skin-on duck breast or lamb chops, for example. In either case, a red wine with some tannin and good acidity is in order. A young Pinot Noir for the duck and Bordeaux for the lamb chops are both natural partners.
If you are having rich meats that are leaner, be on the lookout for a wine with some age on it. The mature wine will have softer tannins. The easiest route to take here is to opt for wines that have seen extensive barrel aging, such as Rioja’s Gran Reservas or Chianti’s Riservas. Both tend to be smoothed by barrel aging and offer a medium-bodied style with bright acidity, another tool to help balance the richness of a dish.
One thing to remember is that there are very few wine and food pairings that are truly bad, most work just fine. Stop worrying and make an educated guess based on these simple pairing guidelines. I’m sure you’ll do just fine!