In a medium size bowl, combine avocado, corn and chives. Season to taste with salt and pepper.
Using a vegetable peeler, make 8 long, thin slices of cucumber by peeling both cucumbers lengthwise. Place 4 slices of cucumber on a flat surface side by side on top of a piece of plastic wrap and pat dry with paper towels.
Place 4 cheese slices over cucumber slices covering most of cucumber. Carefully spread a thin layer of avocado mixture over cheese. Using the plastic wrap as an aid, lift ends of the cucumber slices up and roll over the filling. Continue rolling up cucumber slices all at once making 4 cucumber rollups at once.
Use a knife to cut through each cucumber rollup at the edges. Place rollups on their side on a serving plate. Secure with toothpicks if needed.
Repeat with remaining cucumber slices, cheese and avocado filling.