In a large pot, heat 1 tablespoon olive oil over medium heat. Add the beans and garlic and season with salt and pepper. Cook, mashing, until warmed, about 5 minutes.
Wrap the tortillas in a paper towel and heat in the microwave for 1 minute. Lay flat on a work surface and divide the beans, avocado and cheese among them; roll up like burritos. Wrap in paper towels and microwave until the cheese is melted, 1 to 2 minutes.
In a large bowl, whisk together the lemon juice and remaining 2 tablespoons olive oil; add the cucumber and jicama and toss. Season with salt and pepper. Serve with the burritos.