Grilled Chicken Greek Salad

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1/3 cup extra-virgin olive oil, plus more, for the grill

2 large boneless, skinless chicken breasts (about 1 1/2 pounds), cut into 1-inch pieces

Kosher salt and freshly ground black pepper

2 tablespoons fresh lemon juice

1/2 teaspoon dried oregano

8 cups chopped romaine lettuce

1 cup sliced cucumber

1/2 cup pitted Kalamata olives

1/4 cup thinly sliced red onion

8 ounces feta cheese, broken into large chunks

2 pieces pita bread, for serving