Preheat grill to high heat using both burners. After preheating, turn the left burner to medium heat (leave right burner on high heat).
Combine first 7 ingredients in a mini food processor; process until finely chopped. With processor on, gradually add water, 1 tablespoon at a time; process until smooth.
Combine jicama, radishes, onions, and oil, tossing to coat. Arrange jicama and radishes on right side of grill rack (over high heat) and onions on the left side of grill rack (over medium heat); grill 4 minutes on each side or until tender and well marked. Remove from grill; cut radishes in half. Sprinkle vegetables with 1/4 teaspoon salt. Keep warm.
Sprinkle chicken with remaining 1/4 teaspoon salt and black pepper. Arrange chicken on right side of grill rack (over high heat); grill 4 minutes on each side or until done. Place 1 chicken thigh on each of 4 plates; divide vegetables evenly among plates. Serve each with about 1/4 cup sauce.
This is so yummy! The chimichurri is fantastic. I used two parts parsley to one part cilantro because that was all I had, but it worked. My husband said that is what parsley is made for. I've never grilled jicama before, and it was great; something new I will try again.
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