Foods with more fiber than sourdough bread include legumes, chia seeds, and berries. annabogush / Getty Images Fiber is a carbohydrate that helps maintain healthy digestion, steady blood sugar levels, and benefits heart health. Black beans, lentils, raspberries, chia seeds, sprouted-grain bread, and avocados provide more fiber than sourdough bread. Gradually add high-fiber foods to your diet and drink plenty of water to meet daily fiber goals and avoid bloating.