Put potatoes in a medium saucepan and cover by 2 in. cold water. Bring to boil over medium-high heat and boil until tender when pierced with a knife tip, 8 to 10 minutes. Drain.
Crush garlic and 3/4 tsp. salt in a mortar and pestle into a paste. Add pimentón and 6 tbsp. oil and stir until well combined.
Cut tuna into 1- by 1 1/2-in. pieces. Combine tuna with half of pimentón oil; stir to coat.
Slice cucumber halves into 1-in.-thick pieces (you'll need 12 pieces). Put cucumbers and potatoes into a medium bowl. Add pepper, remaining 2 tbsp. oil, and remaining 1/4 tsp. salt; toss to coat.
Heat a grill to high (450° to 550°). Thread cucumber, potatoes, and tuna onto 4 metal skewers. Wipe grill with oiled paper towels. Grill skewers, covered, 30 seconds per side (2 minutes total), or until tuna is mostly opaque but still pink in center (it will continue to cook off the grill). Serve with lemon wedges and remaining pimentón oil for drizzling.