I just walked the LT stoveless last month, and wouldn’t do it any other way. I’m planning on doing the AT within the next year or two, and will probably do it the same way. (And I say this as someone who works in kitchens and who’s an avid cook at home!) My initial plan was to hike 100% raw (a la Ray Jardine), but soon realized that this would be too expensive, and that I didn’t really want to go three weeks without chocolate 
Here’s my basic method: I carried a good watertight tupperware-type container (lock-n-lock brand), I think it has about a 1.5 cup capacity. Before I left, I made my own muesli-style cereal, mostly with bulk ingredients found at a local discount food store. I don’t have a specific recipe, but ingredients are as follows: rolled oats, hemp seed protein powder (this was pretty pricey, but it’s the only protein supplement I’ve found that agrees with me), seeds (sesame, sunflower, pepitas), nuts (almonds, cashews, walnuts, hazelnuts), dried fruit (dates, figs, apricots), and shredded unsweetened coconut (for me, the coconut flavor is key!). I put this through the food processor to obtain a fairly fine consistency, somewhere between stone-ground flour and quick-oats. I’d guess my ingredient ratio was about 4:2:1 - oats:fruit/nuts:hempseed. But that’s just a guess.
In the evening, I’d put as much muesli as I thought I’d want in the morning into my tupperware, add enough water to cover, and let it soak overnight. When I woke up, instant breakfast! Most days, I’d mix up another batch after breakfast to have for supper. The single dish/single utensil method made clean-up real easy. I only washed the tupperware every couple of days, since it always had the same stuff in it. (If I left it for too long, it would start to smell pretty yogurt-y like the leftovers were starting to ferment.)
As others have mentioned, I didn’t worry too much about meals and just ate whenever I was feeling hungry (or tired or cold). I carried fresh apples and carrots with my the whole time. My thoughts on carrying fresh produce: the extra weight is WATER, which the body needs anyway, and which we’re carrying plenty of anyway. There were plenty of times when I was really glad I had my apples because they did a good job keeping me hydrated, unlike salty and/or dehydrated snacks. When I stopped for groceries, I made a point of picking up other fresh produce as well (radishes, snap peas, avocados, green beans) that I could eat within a day or two.
I always had a variety of nuts and seeds with me, usually raw almonds, cashews, and sunflower seeds. I also had some plain rolled oats for times when I was sick of muesli–all of this I usually made a point of soaking for at least a few hours.
After about a week and a half, I ditched the raw foods diet because I started inheriting food from other hikers–who can say no to hand-outs and trail magic! I ate A LOT of Clif Bars, to which I’m completely addicted (seems most people are looking for a way to get rid of em, so I readily gobbled up everyone else’s). Other standards: peanut butter, of course–on pitas or crackers or tortillas, or just by the sporkful. If I weren’t a vegetarian, I’m sure I would have brought tuna/salmon/sardines/jerky/etc. I did eat my first-ever can of sardines on the trail–that was quite an experience! One guy gave me honey packets (just like ketchup packets) and those were quite a treat. Oh, and perhaps my favorite thing of all: SEAWEED! I went through probably 3 packages of various kinds of seaweed from the coop–dulse, laver, and nori (the kind used for sushi wrappers). Salty, chewy satisfaction, plus it contains lots of minerals.
If I were hiking later in the season with nights below freezing, I might consider bringing a stove just to heat up water for tea and a hot water bottle. But remember: there are other people on the trail and if you ask nice you can probably borrow a stove for a few minutes if you’re truly desperate (just give them something in return for the fuel you use).
I heard stories of other hikers who sent their stove home and then started going crazy without it, but I was perfectly happy. Mind you, everything I was eating was more or less stuff that I would eat at home anyway. Oats and apples are pretty much my staple foods on the trail or off, fleshed out with some nuts and chocolate and other calorie-dense stuff. Plus, when I got home, making omelets and soup and french toast and grilled cheeses and pizza etc etc felt super-exciting!
Hand Me Down