Grilled corn tortillas, shredded lettuce, and cheese provide a tasty bed for this chicken, but it's the Tex-Mex marinade with lots of lime juice that really completes the dish.
Place the chicken thighs in a gallon-size zip-lock bag and add all the marinade ingredients. Press the air out of the bag and seal it. Turn the bag to thoroughly coat the chicken, place it in a bowl, and refrigerate it for at least 4 hours (preferably overnight), turning the bag occasionally. Remove the meat from the refrigerator 20 minutes before you want to start grilling.
Prepare a charcoal fire or set a gas grill to medium-high, close the lid, and heat until hot -- about 10 to 15 minutes.
Remove the thighs from the bag and discard the marinade. Grill the chicken until it's no longer pink inside, about 4 to 5 minutes per side on a gas grill. Transfer the chicken to a cutting board and let it rest for about 5 minutes before cutting it into 1/2-inch strips.
Lightly brush both sides of the tortillas with vegetable oil. Grill them on each side until they turn slightly brown, about 1 minute on a gas grill. Before removing the tortillas from the grill, sprinkle each one with 1 tablespoon of cheese.
To serve, layer the tortillas with shredded lettuce, the chicken strips, the remaining cheese, and any additional toppings. Serves 6 to 8.