What kind of meats? I've eliminated red meat from my diet but I refuse to give up tasty birds like chicken and turkey. They provide more protein and less saturated fat/cholesterol than beef.
Also, not all saturated fats appear to be bad:
Food Myths Debunked: All Saturated Fats Raise Blood Cholesterol / Nutrition / Healthy Eating
As you can see, consuming some types of saturated fat can raise blood cholesterol levels while consuming other types can improve the ratios of blood cholesterol. It’s interesting to note that the main saturated fat of animal products has little effect on blood cholesterol levels. Good news for meat lovers!
As far as "cancer" is concerned, I'm assuming that it's linked to body acidity? Isn't body acidity controlled through eating a lot of fruits and vegetables? So I'm thinking that eating healthy, lean meats in moderation along with fruits and veggies is fine.
I have a couple of friends who are diehard PETA members. They tried to give up all animal protein a few years ago and ate strictly vegetarian. Their attempt failed because it was just too difficult. No, the only way I'm going to give up chicken is when they have to pry it from my cold dead hands.
Great website you're managing there! Thankfully I'm not diabetic but I've becoming more aware of the glycemic index recently and have been using it for my weight loss efforts.