Have ready 7 to 9 skewers. If using wooden or bamboo skewers, soak them in water for at least 30 minutes before grilling.
In a bowl, combine the garlic, salt, pepper, oregano, oil and lime juice and stir to form a paste.
Cut chicken breasts into 1-inch chunks and place in a glass container with a lid. Pour the marinade over the chicken and stir to combine. Cover the chicken and refrigerate for a minimum of 2 hours, up to overnight.
Prepare a grill for direct cooking over medium heat (325 to 375F [170 to 190C]). Depending on the type of grill this may take 15 to 20 minutes.
Remove the chicken from the refrigerator and thread it onto the skewers, spreading each piece as flat as possible and leaving a very small space between each piece.
Once the grill is hot, brush the cooking grates clean, if necessary (to prevent sticking). Grill the kebabs over direct medium heat, keeping the lid closed as much as possible, until the chicken is firm to the touch and no longer pink in the center, 8 to 10 minutes total, turning once or twice during cooking. Take care not overcook.
Remove from the grill and serve immediately.