Despite its under-the-hood complexity, bread contains just a few fundamental ingredients. If you have flour, water and yeast -- even salt is optional -- you can make a workable loaf of bread. If you're out of yeast, or have an allergy to yeasts, this complicates matters. Still, there are many breads made without yeast. Some are quick and simple, while others are quite complex.
If your goal is simply to eliminate mainstream, mass-produced yeast, that's a relatively simple goal. It only became available in the late 19th century, and most breads produced before that time relied on wild yeasts. You culture the yeasts by setting out a wet batter to trap spores, then feeding it regularly as the yeasts begin to grow and reproduce. The resulting sourdough starter gains its distinctive tang from lactic acid bacteria that form colonies in the dough, making it acidic enough to deter spoilage-promoting rival bacteria. Sourdough makes more interesting loaves than bland, workmanlike commercial yeast; but the loaves are still yeast-raised and unsuitable for anyone with a yeast-restricted diet.
A different form of fermentation is responsible for salt-rising bread, an old-time technique that's still preserved in some parts of the country. Traditional recipes vary but typically use a starter based on scalded milk or water that's had potatoes boiled in it. After sitting overnight in a very warm place, at 90 to 100 degrees Fahrenheit, bacterial fermentation makes the mixture start to froth. Add flour to make a thick batter, and over a period of hours the starter will rise and become foamy, producing a penetrating and distinctive odor. Kneading flour, salt, and softened butter into the starter produces your finished dough, which makes a dense, sweet and slightly cheesy-tasting loaf. The "salt-rising" name is a mystery, since the bread is no saltier than other varieties.
Although tasty, salt-rising bread is notoriously failure prone and time consuming. Quick breads are exactly the opposite, using reliable dry leaveners and easy, quick mixing. Traditional Irish soda bread is a classic example, combining baking soda with acidic buttermilk to raise the bread. Other recipes use baking powder, which is even easier for the baker because it contains its own acidic ingredients and doesn't need sour milk or butter milk. Quick breads can be simple and Spartan, or rich with added ingredients, but in either case are yeast free.
Flat breads are in some ways the simplest unleavened breads. They don't need workaround techniques to produce a light crumb, because they're used primarily for wrapping and dipping rather than slicing. Basic flat breads such as flour tortillas are chewy and uncomplicated, but their counterparts in other cultures can be more sophisticated. Many Indian flatbreads develop a characteristically pita-like pocket as the dough's moisture turns to steam under intense heat, forcing the sides apart. Other Indian flatbreads, such as parathas, are brushed with oil or butter and then repeatedly rolled and flattened to create a delicate, flaky texture. If you're avoiding yeast, note that many flatbreads -- including pita -- often contain yeast. If you're buying your bread, check the label to be certain.