I have been a fan of Mark's Daily Apple, written by Mark Sisson, who also wrote The Primal Blueprint and a number of (decent) cookbooks and other guides, for a while. He talks about living "primally in a modern world." Others call eating in this way "Paleo." Sisson brings his "Primal" concept a bit further than just eating how our ancestors ate in the millions of years before we began farming and eating complex carbohydrates in earnest (10,000 years ago). He also discusses lifestyle tweaks that are more in line with how our bodies were developed to perform-- pre-industrial revolution and pre-agricultural revolution.
Recently, I have read more on the topic; namely, the food aspect and all of the advantages of eating low-carb enough to be fat adapted (which Sisson only touches upon- it's not his main focus). In the excellent book Why We Get Fat, Gary Taubes lays out an argument of why we should all eat low enough carb to be keto-adapted (fat adapted). His book is well researched while still being easy to read, and he lays out the counterarguments one by one and knocks them down. I highly recommend it to anyone.
Phinney and Volek helped re-write the Atkins book (which I've yet to read), and also wrote The Art and Science of Low Carbohydrate Living and The Art and Science of Low Carbohydrate Performance. The Art and Science of Low Carbohydrate Performance is concise and makes a clear and researched argument that athletes can perform while keto-adapted (aka "in nutritional ketosis" or just "in ketosis"). They even include some recipes for how to eat more fat. They say that you need to eat 1 g of protein per pound of lean body weight (approximately) - which is moderate protein, and about as much as most people eat anyway- but then to lower your carbohydrate consumption and replace it with fat. What is novel about their approach is that they say to test your blood ketones to tweak your carbohydrate consumption to stay in ketosis (free meter available here though the sticks are pricey-- they argue the urine sticks aren't accurate enough). They say most people end up in ketosis when eating between 20-50 carbohydrates per day, or many more or less, and it can change as your body changes. But this leaves 70% of your daily food intake as fat (if protein stays the same and carbs stay the same). The charm of being keto-adapted is that it trains your body to run on fat as fuel and if you need more fuel (for example, while training), your body keeps much in reserves. It takes about 14 days, they say, to train your body to stop running on carbs and start running on fat, and you may not feel great during this time, but you will feel great after, as long as you have a "well formulated low carb diet," which means your percentages are right and you are getting enough salt. I am partially through The Art and Science of Low Carbohydrate Living and it is a lot denser, but I think it was written for a more scientific audience. It touches on the same points, but also expands on how all of our bodies know how to run on fat but we just need to give them enough of the right fats and not clog them up with carbs. One of their major points is to make sure you are eating enough fat-- low carb doesn't mean high protein, it means high fat.
The GAPS Diet (I discussed it here as well) is a similar high fat diet targeted at healing people and gut health. It removes grains and requires "real" whole foods, but doesn't require keto-adaptation.
From what I have read, I am thinking that keto-adaptation may be the key to getting the body to normalize (in health- as shown in weight management and standard health markers like blood pressure, cholesterol, general energy levels, etc.). The local Bay Area company Jump Start MD uses Ketosis for their clients. Medifast uses Ketosis with calorie restriction (though they don't use real foods as their primary tool).
There are a few websites in addition to Mark's Daily Apple which discuss eating without grains to improve your health (here is a post of his that relates to Ketosis). In addition to all the Paleo sites, Athlete.io (formerly Dangerously Hardcore) is a wealth of well-researched information relating to using nutritional ketosis to burn fat more efficiently, with and without resistance training. Jimmy Moore's Livin' La Vida Low Carb is actually quite informative, but in more of a storytelling way with himself as the story. Moore also writes on Carb Smart (like this post on advantages of being keto-adapted). I would be remiss if I didn't mention Martin Berkhan and Leangains and his discussion of food timing and its impact (Keifer at Athlete.io discusses this at length as well - like in this post about why you should skip breakfast).
In conclusion, I think this way of eating~ low carb and high fat, with real foods of course as the staple foods, seems to be what our bodies respond best to. These books and websites reference scientific studies and personal experience. I recommend further reading for all of us.