Preheat a gas grill to medium (about 400°F) on one side, or push hot coals to one side of a charcoal grill. Place a disposable aluminum pan directly on the burner on lit side of the gas grill, and add soaked wood chips to pan, or scatter wood chips on hot coals of charcoal grill; heat until wood chips are smoking.
Place potatoes in a disposable aluminum pan; place on grates over unlit side of grill. Grill, covered, 20 minutes. Rotate pan halfway; grill, covered, until potatoes are tender, 20 to 30 more minutes. Cut potatoes into 3/4-inch pieces. Place in a large bowl; add radishes.
Whisk together oil and remaining ingredients in a small bowl. Pour dressing over potatoes and radishes; toss to coat.