Rinse and dry tri-tip. Rub with 1 tbsp. olive oil. Whisk together 1 tsp. cumin, dried oregano, 1 1/4 tsp. salt, and 1/2 tsp. pepper; massage into tri-tip. Let sit 30 minutes.
Make sauce: In a blender, whirl juices with remaining cumin, the fresh oregano, the garlic, 1/2 tsp. salt, 1/2 tsp. pepper, and remaining olive oil. Season to taste with additional salt and pepper.
Prepare a charcoal or gas grill for medium heat (350° to 450°; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds), leaving a cleared area if using charcoal. Grill tri-tip, turning to brown evenly and, on charcoal, moving it to wherever heat is medium (see "Our Live-Fire Grilling Secrets," below), until a thermometer inserted in center reads 125° to 130° for medium-rare, 25 to 30 minutes. Let meat rest 15 minutes, then cut across the grain into thin, slanting slices. Serve with sauce.
Our Live-Fire Grilling Secrets: Gas is no big deal, because the heat stays steady. Charcoal or hardwood is a different game. Here are a few hints for staying on top of yours.
Keep the Firegrate Ash-free. A big pileup blocks the airflow, choking the fire so it can't burn.
Get to Know Your Coals and How They Burn. Start by following the directions on the package (seriously). Kingsford's new grooved coals, for instance, burn hotter and longer than the smooth kind, so a single layer of coals puts out a lot of heat. Hardwood charcoal pops and sizzles and burns unevenly, but gives great flavor.
Create an Emergency Cool Spot. If your grill is big enough, leave an area of the firegrate free of coals. You'll typically use this cool spot at least once whenever you grill.
Measure the Heat Constantly. Use your grill's thermometer or, if it doesn't have one, your hand. Low heat (250° to 350°) means you can keep your hand 5 in. above the cooking grate for 8 to 10 seconds before you have to yank it away. For medium heat (350° to 450°), you can keep your hand steady for 5 to 7 seconds; for high (450° to 550°), 2 to 4 seconds; for very high (550° to 650°), 1 to 2 seconds.
Move Your Food Around. Once you've identified which spot on the grill has the desired heat, get your food over there (often the emergency spot will have just the temperature you need, if only for a little while). You may have to move food several times before it has finished cooking. The exception: food that cooks in a flash over a super-hot fire.
Manipulate the Fire. Need it hotter? Push coals together or throw on a few more. Cooler? Separate the coals or cover the grill with the lid. Close lid vents to cool it off even more. (Don't close vents under the firegrate, though, or the fire will go out.)
Note: Nutritional analysis is per serving.