Place the cream cheese in the freezer while you prepare the other ingredients—this ensures easier slicing later!
In a small bowl, whisk together the olive oil, vinegar, red peppers, green onions, garlic, sugar, oregano, salt, and black pepper until well combined.
Slice the cheddar and mozzarella into squares about ¼ inch thick. Remove the cream cheese from the freezer and slice into similar-sized squares.
Arrange the cheese slices in an 11×7-inch baking dish, alternating layers of mozzarella, cheddar, and cream cheese until the dish is full.
Give the marinade a quick whisk to recombine, then pour it evenly over the cheese layers.
Cover the dish tightly with plastic wrap and refrigerate for at least 5 hours or overnight to let the flavors meld.
Serve the marinated cheese with crackers, pretzels, or toasted bread, and watch it disappear!