In a large pot, drizzle olive oil over medium heat.
Once thoroughly heated, add the onion, celery, and garlic. Cook, stirring often, until softened, about 5 minutes.
Add the diced tomatoes, broth or water, basil, bay leaves, oregano, salt and pepper. Bring to a boil, then reduce heat to a simmer and cook until the tomatoes are softened and starting to fall apart, 20-30 minutes.
Remove the bay leaves, then puree the soup using an immersion or stand blender until smooth. If using a stand blender, carefully transfer the soup back to the pot after blending.
Return the soup to a simmer, then stir in the orzo. Cook stirring often (especially the bottom and sides) until tender, about 5-10 minutes.
Remove from heat and stir in the yogurt. Serve immedately topped with crumbled or grated cheese and additional basil.