Stir the canola oil, root beer, Worcestershire sauce, teriyaki sauce, steak sauce, garlic salt, onion salt, black pepper, basil, thyme, and rosemary together in a bowl.
Pierce the steaks in several places on both sides; arrange in a single layer in a shallow non-reactive dish.
Pour the marinade over the steaks, flipping the meat to coat both sides.
Allow to marinate in refrigerator 3 to 4 hours, flipping meat once halfway through.
Preheat an outdoor grill for medium-high heat, and lightly oil the grate.
Remove the steaks from the marinade and shake off excess moisture; discard the remaining marinade.
Cook the steaks until they are beginning to firm, and are hot and slightly pink in the center, 5 to 7 minutes per side.
An instant-read thermometer inserted into the center should read 140 degrees F (60 degrees C) for medium.