This is perhaps the most common reaction I get when telling people about the low carb diet (LCD). The belief that saturated fat will lead to heart attacks is well ingrained in the American psyche. (It's also illogical. Obesity leads to heart attacks, so getting lean reduces heart attacks, but a diet designed to make you lean also leads to heart attacks? You can't have it both ways). Just because you believe something doesn't make it true. There has been a fair amount of research recently on the effects of the LCD (see below).
So no, in fact, I won't have a heart attack because of following an LCD. But I might very well have had one following a traditional high-carb, low fat diet.
Recent studies in major journals all confirm that the LCD improves cardiovascular health. This blog post by a Ph.D. in medical biochemistry links to and summarizes the results of three major studies appearing in the Annals of Internal Medicine, The New England Journal of Medicine, and The Journal of the American Medical Association. All show that an LCD lowers risk factors for cardiovascular disease, such as high blood pressure, high blood sugars, and high cholesterol. Plus click on the review article listed below, which concludes basically the same thing. The LCD is a heart healthy diet.
A 2007 review article in the American Journal of Clinical Nutrition concluded that LCDs "lead to appetite reduction, weight loss, and improvement in surrogate markers of cardiovascular disease." Further, it concluded that LCDs improved blood sugar levels in both healthy people and those suffering from type 2 diabetes.