Meanwhile, in a large bowl, mix the beef, cumin, garlic powder, and oregano. Form the mixture into 1/2-inch-thick patties; they should be about the same diameter as your buns. Sprinkle the patties with salt and pepper.
Grill the patties, turning once, until browned and cooked through (to 165 Degrees F), about 8 minutes. During the last minute of cooking, grill the buns until toasted, about 30 seconds.
Spread the dip on the bottom buns, top with the burgers, and sprinkle with the cheese. Arrange the cucumber and tomato slices on top. Sandwich with the top buns.