Cut off peel and white pith from 3 oranges with a sharp knife.
Cut peeled fruit in half lengthwise, then cut crosswise into 1/4-inch-thick slices.
Cut remaining orange in half crosswise, then cut each half into 6 to 8 wedges for squeezing.
Cut jicama wedges and cucumber crosswise into 1/4-inch-thick slices.
Squeeze juice from 4 to 6 orange wedges into a bowl, then add jicama, cucumber, and salt to taste and toss well. Season orange slices with salt and arrange on a platter with jicama and cucumber. Lightly sprinkle with cayenne.
Serve with remaining orange wedges.
Oranges, jicama, and cucumber can be cut 3 hours ahead and chilled separately in sealed plastic bags.
