Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling, stir in the linguine and return to a boil. Cook the pasta uncovered, stirring occasionally until the pasta has cooked through but is still firm to the bite, about 11 minutes. Drain well in a colander set in the sink.
Meanwhile, heat the olive oil in a large saucepan over medium heat. Stir in the garlic and cook until softened, about 3 minutes. Stir in the tomatoes and bring to a simmer. Cook 1 minute before adding the spinach. Cook and stir 5 minutes; season to taste with salt and pepper. Stir the drained linguine into the tomato sauce before serving.