Heat a grill to medium high. Lightly brush the grates with olive oil. Slice the potatoes 1/4 to 1/2 inch thick and toss with 2 tablespoons olive oil, 1/2 teaspoon salt and 1/4 teaspoon pepper. Spread on a sheet of heavy-duty foil. Bring the ends of the foil together and seal into a flat packet. Grill, flipping the packet halfway through, until the potatoes are tender and lightly browned, 15 to 20 minutes.
Meanwhile, put the turkey, remaining 1 tablespoon olive oil, lemon slices, garlic and oregano sprigs in a large resealable plastic bag. Massage the marinade into the meat, about 3 minutes. Remove the turkey and season with salt and pepper. Grill, turning once, until cooked through, 6 to 8 minutes. Brush the tomato halves with olive oil and grill, turning once, until lightly charred, about 3 minutes.
Carefully open the foil packet and toss the potatoes with the chopped oregano; top with the feta. Cut the tomatoes into wedges and halve each turkey cutlet. Divide the potatoes, tomatoes and turkey among plates, along with the grape leaves, hummus and lemon wedges. Drizzle with olive oil.