For dressing, in a screw-top jar combine olive oil, vinegar, lime juice, and sugar. Cover and shake well. Chill for 2 to 24 hours.
Peel and section the oranges over a large mixing bowl to catch the juice. Add orange sections, pineapple, and apple slices to the bowl. Toss to coat all of the fruit with orange juice. Cover and chill for 2 to 24 hours.
To serve, line 6 salad plates with romaine leaves. Top with shredded lettuce. Add banana slices to the fruit mixture; toss to coat bananas with juice. Drain the fruit mixture. Divide the fruit mixture, beets, and jicama among the salad plates. Sprinkle each serving with pine nuts or peanuts and pomegranate seeds or red onion. Shake the salad dressing and pour some over each salad. Makes 6 side-dish servings.