I wouldn't do a traditional low-carb diet because I get more migraines when I eat less carbs. (In fact, it took me a long time to realize that the reason I always felt bad when trying to eat "healthy" is that I need to eat a larger percentage of carbs than the average person.
I do think that eating low carb can be healthy, but as with anything else you have to eat mindfully. "Low carb" diets don't usually count vegetables as carbs (although my body does), so you can certainly eat a variety of healthy foods on a low carb diet. And if you gorge on bacon or steaks or whatever it's just as unhealthy as eating just grapefruit or cabbage soup.