Wash and peel turnips using a vegetable peeler and a steady hand. Cut in half or in quarters, so that they are similar in size to the radishes. Wash the radishes.
Place veggies in a bowl and add the olive oil, salt, a sprinkle of pepper, and a sprinkle of garlic powder, if you’d like. Mix well.
Dump the veggies onto a large piece of tin foil (If you aren’t using ‘heavy duty’ tin foil, I would suggest laying down two pieces, one on top of the other, so that your pocket doesn’t break open on the grill.) and fold together the sides to make a sealed pouch.
Place on the grill, over indirect heat, and let grill for 10 to 15 minutes, depending on the size of your veggies. (If you don’t have a grill, these are just as good in a 400 degree oven, for closer to 20 minutes.) Serve.
Careful when opening the pouch! There will be steam and it will be hot.