Make marinade:

In a small saucepan bring marinade ingredients to a boil, stirring until sugar is dissolved, and cool.

Preheat oven to 400 degrees. Wrap beets tightly in foil and roast in middle of oven 1 to 1 1/2 hours, or until tender. In a large saucepan of boiling salted water cook carrots until crisp tender, 6 to 7 minutes, and transfer with a slotted spoon to a bowl. Unwrap beets carefully and cool. Slip off beet skins and slice beets, transferring to another bowl. Divide marinade between beets and carrots, combining well, and marinate, covered and chilled, at least 4 hours.

In saucepan of boiling carrot cooking water cook beans until crisp tender, 5 to 6 minutes, and drain in a colander. Rinse beans well under cold water and pat dry between paper towels. Vegetables may be prepared up to this point 1 day ahead and chilled, covered.

Make dressing: In a bowl whisk together mustard, vinegar, water, sugar, and salt and add oil in a stream, whisking until dressing is emulsified. Whisk in dill.

Just before serving, toss beans in a bowl with 2 tablespoons dressing and drain beets and carrots. Arrange vegetables decoratively on a platter and drizzle with remaining dressing.