Combine oil, lemon juice, oregano, and 1 teaspoon salt and 1/2 teaspoon pepper. Reserve and refrigerate about 1/4 cup of marinade. Transfer remaining marinade to a zip-top bag and add chicken and toss to coat. Refrigerate 8 hours (or up to overnight).
Heat a grill or grill pan to medium-high. Clean and lightly oil hot grill. Thread some skewers with chicken, others with tomatoes, and others with bread. Season with salt and pepper. Grill bread skewers for about 3 minutes, tomato skewers for 3 to 6 minutes, and chicken skewers for 12 to 15 minutes, turning all skewers frequently and brushing with reserved marinade.