Preheat the oven to 325 degrees.
Line a 4 x 8 bread pan with parchment paper.
Sift the flour, baking soda, baking powder, salt, cinnamon, and allspice in a large mixing bowl.
Mix the sugar, eggs, vanilla, Greek yogurt, and vegetable oil in a separate mixing bowl. Pour the mixture into the dry ingredients, and mix well.
Fold the grated zucchini into the batter.
Evenly distribute the batter into the lined bread pans.
Bake for 45-50 minutes. Allow the zucchini bread to cool for 15-20 minutes before slicing.