Preheat the grill, setting the burners on high. Oil the grate.
Thinly slice 2 radishes on a mandoline and set aside.
Halve the remaining radishes. Place in a bowl and combine with the onion and oil. Season with salt and pepper.
Grill the vegetables directly on the grate or in a barbecue wok for 5 minutes or until tender. Place in a bowl. Add the arugula, cherries and reserved radish slices. Drizzle with olive oil and adjust the seasoning.
Serve with spiced chicken skewers (see recipe).
Depending on the width of your grill’s grate, you might want to cook the vegetables in a grill wok.