If using wooden/bamboo skewers, soak them in water for at least 30 minutes to prevent burning.
Cut chicken into 1 to 1.5-inch pieces for even cooking.
In a large bowl, stir together pesto, olive oil, lemon juice, salt, and black pepper. Add chicken and toss until well coated.
Marinate at least 15 minutes (up to 4 hours in the refrigerator). Avoid marinating longer than 8 hours for best texture.
Thread chicken and cherry tomatoes onto skewers, alternating pieces. Leave a little space between pieces so heat can circulate.
Preheat grill to medium-high heat (around 400°F / 205°C). Oil the grates or grill pan lightly.
Grill kebabs 4–5 minutes on the first side, then flip and grill 4–5 minutes more, until the chicken is golden with a few charred spots and reaches 165°F (74°C) in the thickest piece.
Remove from heat and rest 2 minutes. Serve warm with rice, salad, pita, or your favorite sides.