Preheat your oven to 350°F (175°C). Grease and flour a 9×5-inch loaf pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, unsweetened cocoa powder, baking soda, baking powder, and salt. Set aside.
In a large bowl, combine the granulated sugar, light brown sugar, and vegetable oil. Beat until well combined.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in the vanilla extract and mashed ripe bananas until fully incorporated.
Add the grated zucchini and mix until evenly distributed.
Alternately add the dry flour mixture and the Greek yogurt to the wet ingredients, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack and let it cool completely before slicing and serving.