Beef eye-round is one of those underappreciated cuts; it is very lean, has the same texture as sirloin, but slightly more beefy flavor. It must not pass medium-rare or it will be tough.
This was an easy preparation. I marinated two 2-inch thick steaks overnight in equal parts of white vinegar and olive oil (about 2 tablespoons each). I salted the meat before grilling. 10/10/20 on the grill: 10 minutes per side on high with the cover up, 20 minutes on the cold side of the grill with the opposite burners set on "low", cover down.