Score a diamond pattern by making shallow diagonal cuts on both sides of beef. Season beef lightly with salt and pepper. For a charcoal grill, place the beef on the grill rack over medium-hot coals. Cover and grill about 10 minutes or until desired doneness. (For a gas grill, preheat grill. Reduce heat to medium-high. Add beef to grill rack. Cover and grill as above.) Remove from grill rack; let stand 10 minutes before slicing.
Meanwhile, in a medium bowl toss together the cucumber, oil, dill, vinegar, 1/8 teaspoon salt, and 1/8 teaspoon pepper.
Thinly slice steak across the grain. Serve beef slices over cucumber mixture. Makes 4 servings.