Preheat oven to 350°F. Spray two 8½ x 4 inch loaf pans with baking spray. Line with parchment paper, allowing it to overhang the long sides by 1–2 inches.
Combine all three types of zest in a small bowl and mix lightly with a fork. Set aside, reserving a small pinch for garnish.
Combine all the citrus juices (lemon, lime, and orange) in a bowl. Divide as follows: place 5 tablespoons in a small saucepan (for the syrup glaze); place 3 tablespoons in a medium bowl (for the icing); set aside the remaining 4 tablespoons for the cake batter.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, and poppy seeds. Set aside.
In a separate bowl, combine Greek yogurt, milk, 4 tablespoons of the citrus juice mixture, and vanilla extract. Whisk together and set aside.
In the bowl of a stand mixer, beat butter and sugar together on medium-high speed until light and fluffy, about 2–3 minutes. Add eggs one at a time, mixing well after each addition.
With the mixer on low, add the flour mixture and the yogurt mixture alternately, beginning and ending with the flour mixture. Mix until just combined. Add the citrus zest (reserving a pinch for garnish) and stir to combine.
Divide batter evenly between the two prepared pans. Place pans on a baking sheet. Bake for 50–60 minutes, or until a skewer inserted into the center comes out clean.
While cakes are still hot, prepare the syrup glaze: add ⅓ cup sugar to the saucepan with the 5 tablespoons of citrus juice. Bring to a boil over medium heat, stirring until sugar dissolves. Remove from heat.
Using a skewer or toothpick, poke holes all over the tops of both hot loaves. Slowly drizzle the hot syrup over both cakes, allowing it to soak in before adding more. Use up all the syrup.
Allow cakes to cool completely in the pans, then lift out using the overhanging parchment paper.
For the icing: whisk 1½ cups of powdered sugar into the bowl with the 3 tablespoons of citrus juice until smooth. Add more powdered sugar as needed to achieve a thick but drizzleable consistency. Drizzle back and forth over the tops of both loaves, letting the icing drip down the sides. Garnish with reserved citrus zest.