Pat dry the salmon fillets with paper towels on a cutting board. Cube the salmon into 1-inch cubes.
Slice the lemons into thin slices, about ¼-inch thin, so you can fold them in half and poke them through a skewer.
In a large bowl, whisk the olive oil with lemon zest, lemon juice, garlic, oregano, salt, and black pepper. Add the cubed salmon to the bowl and toss to coat it all with the marinade. Cover with a lid or plastic wrap and refrigerate for 15 minutes.
Place the wooden skewers on a baking sheet and fill with water. Soak them while the salmon is marinating.
Assemble:
Slide the marinated salmon cubes through the skewers, alternating with the lemon slices. Place them on a baking sheet or platter as you go.
Grill:
Preheat the grill to high heat. Once hot, reduce the heat to medium-high (between 375 to 400F). Brush the grill with oil.
Place the skewers on top of the grill, in the opposite direction of the grates. Grill with the lid, closed for 5 minutes, open the lid, carefully flip them over, and finish grilling them with the lid open for another 5 minutes until the salmon has turned a nice pinkish-white color. Remove from the grill.
Notes
If your grill grates are wide, you can use a grilling pan on top of the grill and cook the salmon skewers on it. Make sure to grease it too. Alternatively, you can use heavy-duty aluminum foil on top of the grill.