Jicama is a tuber that is similar in texture to a potato and in flavor to an apple. They are best when smooth and firm, with thin skins.
Combine tomatoes, vinegar, oil, and salt and pepper to taste in a glass jar. Shake well and refrigerate overnight.
Peel jicama by cutting a thin slice off the top and bottom and peeling all sides with a sharp knife. Cut into slices about 3/8 inch thick. Using assorted star-shaped cookie cutters, cut jicama slices into stars. Cover with a damp paper towel until ready to use.
Wash cucumber well; do not peel. Cut off tips and, using a vegetable peeler, cut into long, thin strips.
Place lettuces in a large bowl or on a platter. Arrange cucumber "stripes" and jicama "stars" on top of lettuce.
Using a fork, mash a little of the diced tomato in the dressing. Shake or mix well, adjust seasonings, and pour over salad.