Most of the answers here are quite good, but I seem to have a slightly different take on this. I'm not low-carb, paleo, or allergic to wheat, but I still find that a low-grain diet has nutritional benefits. To me, grains are mostly a "filler" food. They're cheap and can help some with satiety, but don't provide the nutrients that many other types of foods can. Eating low-grain has allowed me to eat more fresh produce, and thus get more nutrients, while still keeping my calories at maintenance-level. Having reduced my grain intake, I can now hit almost all my micronutrient needs without any kind of supplementation. There's a lot of fighting about whether grains are good or bad for you, and a lot of scientific studies that are riddled with flaws, but I won't even go into that. My personal take on it is that grains aren't inherently bad (barring allergies or blood sugar concerns)- they've just become too large of a factor in our diet, and they've pushed out better (more nutrient-dense) food options, especially produce.
In response to your original question, there's a ton of food that you can eat other than bread or other grains. Depends on how restrictive you want to be and the reasons you want to do it. Like I said, I don't eat low-carb and I don't think grains are inherently bad, so I have a lot of flexibility. I just try to eat multiple kinds of produce for every meal, and I sub veggies and greens in place of most grain-heavy meals. For example, I might have eggs with peppers, onions, and tomatoes in the morning, a big salad with dark leafy greens, more fruit/veg on top, plus something filling (dressing, nuts, cheese, meat, etc.), and dinner is a toss-up. Common dinners include meat or fish with roasted seasonal vegetables, vegetable soups or stews, vegetarian burgers (made with quinoa or some type of bean), lentils, etc. I kill pasta cravings with roasted vegetables covered in marinara sauce. I eat fruit like a beast. Once you get to thinking about it, grains are only a very tiny portion of the types of food out there, we've just become over reliant on them.
I also still eat some grains, because as I said above, I don't think they're inherently bad. I just keep it to 1-2 servings a day at most, and I get the rest of my carbs through fruits and vegetables (and some dairy/nuts/etc). Grains for me are a side-dish or add-on rather than the backbone of most meals. For awhile, I did the primal eating thing and didn't eat any grains, but I found it didn't really have any additional benefits (versus low-grain eating) and it felt overly restrictive, and too expensive. So I went back to eating 1-2 servings of grains a day, and I'd recommend a lower-grain higher-produce diet to almost anyone.
Also, two notes because these always seem to come up in grain discussions:
1. Some grains are truly whole grains, but most bread/cereal/pasta products labelled "whole grain" in the store are loaded with crap. It's really difficult to find bread or cereal that doesn't have high fructose corn syrup or a blend of whole and processed grains. That's why oats and rice often have a better reputation in the health world- they're easier to find "whole". You can certainly make your own 100% whole grain bread/pasta or find it in a store or bakery with some effort, and either of those options are arguably as healthy as oats. Grain quality does matter, though it often gets ignored in the are-grains-good-or-bad conversation.
2. Carbs are not synonymous with grains. This is a personal pet peeve of mine. Carbs are a macronutrient that can be found in almost any whole food except meat- fruits, vegetables, nuts, dairy, etc. all have carbs. Different sources of carbs can have very different effects on the body. There's a lot of carb-hate going on in the fitness world right now and I think a lot of it stems from the annoying tendency to equate carbs with grains. The two are different. Specifying which one you mean when talking about healthiness and effects on the body can clear up a lot of confusion. Soap box rant over.