Heat a large skillet over medium-high heat and add the olive oil. When the oil is hot, add the chicken and sprinkle it with the oregano, salt, and pepper. Sauté the chicken until cooked through and no longer pink inside, 8 to 10 minutes. Transfer the chicken to a plate to cool.
Place the ingredients for the tsatsiki sauce in a medium bowl and stir to combine. Refrigerate until ready to serve.
Combine all the ingredients for the salad in a large bowl and toss. Taste and season with more salt, pepper, or lemon juice, if needed.
To assemble the wraps, drizzle some tsatsiki over the center of a tortilla, then top with some salad, some chicken, and a drizzle of more tsatsiki sauce. Roll up the wraps, folding in the ends halfway to contain the filling. Cut the wraps in half to serve.