In a small bowl combine daikon, green onions, vinegar, jalapeno pepper, sugar, and sesame oil. In another small bowl combine shredded carrot, cucumber strips, cilantro, and soy sauce. Cover both mixtures; refrigerate for 2 to 24 hours, stirring once. Drain both mixtures.
Pour the warm water into a pie plate. Carefully dip rice papers into water, one at a time. Place papers, not touching, on clean dry kitchen towels. Let soften for a few minutes until pliable.
Place 1/4 cup shredded lettuce on each rice paper near one edge. Place about 1 rounded tablespoon of each vegetable mixture on the lettuce. Fold in the ends. Beginning at that edge, tightly roll the rice paper. Place, seam side down, on a plate. Cover with a damp towel. Repeat with the remaining fillings and papers. Cover and refrigerate up to 2 hours.
To serve, cut each roll in half crosswise on a diagonal to make 12 pieces. Makes 12 appetizer servings.