Heat a grill or grill pan until very hot. Peel cucumbers in lengthwise strips, leaving thin strips of skin between peelings. Slice cucumbers into very thin rounds. Place cucumbers in a bowl with rice-wine vinegar, 3 tablespoons dill, sugar, and 3/4 teaspoon salt. Toss to combine, and set aside.
Place salmon on a tray, sprinkle with remaining 1/4 teaspoon salt and pepper, and coat with remaining dill. Transfer to hot grill, and cook several minutes, until nicely browned. Turn over, and continue grilling until just cooked through. Salmon will look flaky when done. Place pickled cucumbers on a serving platter, top with salmon fillets, and serve.