Using a spiralizer or mandoline, make zucchini noodles and set them aside. (You can also slice the zucchini thinly, if you don’t have the gadgets for making zucchini noodles.)
Heat a large skillet over medium-high heat. Add 2 tablespoons of the olive oil and the chicken. Cook until golden brown, stirring occasionally. Add the cumin, 1/2 teaspoon of the salt, and the ground black pepper. Cook and stir 1 more minute. Transfer to a bowl and set aside.
Return the skillet to the stove on medium-high heat. Add the remaining 2 tablespoons oil, zucchini noodles, and tomatoes. Sauté for 1 minute. Remove from the heat and add the cooked chicken, remaining salt, jalapeños, lime juice, and cilantro. Stir gently and serve hot.