The brain is actually made out of fats (and cholesterol), it will in fact preferentially burn fats in the forms of ketones over glucose if both are present (experiments were done on this way back).
Where you get into problems is if you have used carbs for fuel for an extended period of time and then switch drastically to a low carb (ketogenic diet), ie. the two week induction phase of Atkins.
You have basically switched gears massively and it takes your brain (and everything else) a few days to switch over from burning carbs to fats and in that time you may feel a little off, how much so depends on the person.
Some people adapt very fast and feel nothing, some people take about a week or so to fully change over and feel a bit "dopey" or sluggish while their body snaps into shape.
Its not just the brain, all your internal organs have large fatty deposits for fuel, of course that is where all your stored energy mainly comes from.
You only have enough carbs for a few hours of work (carry about 100 g in the liver, 200-300 in the muscles), a normal
body weight person (16-18% BF), can live for weeks on body fat alone, every pound you carry has enough energy for 1-2 days.
Just think, if you really need carbs to live and for your brain to function would you not have evolved to actually store them as energy. If you could not depend on fat alone it would be pretty pointless to store it.
The real problem is with people on pseudo-
low carb diets. They never eat enough to go into ketosis, but don't eat enough for the body to get its necessary energy from carbs either, so its constantly switching back and forth and never really working well.
I have had lots of friends have success with low carb diets, but have also seen people fail at them horribly. These are in general the people who like to cheat, sneaking a muffin, break or can of coke, or whatever and thus they never get past the above mention rough patch and thus low carb for them is a horrible process.
But this isn't the fault of the diet, it isn't being applied properly. It like complaining that curls don't grow biceps, and then using form which relies heavily on the hips and/or back to swing the weight up.
-Cliff