This is a loaded question! Everyone has such strong opinions. Die-hard omnivores like me feel like no matter what today's nutritional scientists say, someday they will realize that some amount of meat IS necessary. But the American Cancer Society and probably a lot of other organizations say it's better to not eat meat. I eat meat, probably way too much of it. I would urge you to ask the opposite question: How little meat can I eat and still feel healthy and happy?
I don't want to offend any vegans, but it sounds like you're not interested in following a restricted diet, otherwise you would already be a vegetarian and not asking how much meat you should get. So my advice is strongly influenced by my mom and Michael Pollan. You should look at what healthy indigenous cultures eat (or ate, since almost no one follows old eating habits anymore). Some, like Eskimoes, ate pretty much only meat and seafood. It's been said that the Mediterranean diet is the healthiest there is and I can tell you that in the Orthodox calendar followed in Greece, about half the year is made up of fast days, spread throughout. So no meat on Wed or Fri and then no meat on major fast seasons, like Lent. While I'm not urging you into Orthodoxy, this has influenced the amount of meat they eat in Greece by a lot and it can inform your decision as well. If you pick two or three days a week you won't eat meat, and then use moderation on the other days, I think it would be a healthy decision and a smart way to limit yourself without feeling deprived.
Another rule of thumb I once heard is to not eat more than 3 or 4 ounces of red meat a week, but boy do I blow that every time I eat a burger!