Season chicken on both sides with salt and pepper. Let stand at room temperature 30 minutes, or refrigerate, covered, up to 2 hours. Meanwhile, combine cucumbers, scallion, jalapeño, vinegar, and sugar. Season with salt.
Heat grill to mediumâ€high; lightly brush grates with vegetable oil. Rub chicken with olive oil; transfer to grill. Cook about 2 minutes, then rotate chicken 45 degrees (to create crosshatch marks) and continue cooking another 2 minutes. Flip and grill until cooked through, 3 to 4 minutes more. Transfer to a platter; let stand 10 minutes. Serve with cucumber relish and mint.