Preheat oven to 350°F. Butter and flour a 10-inch bundt cake pan.
In a medium bowl, combine the flour, baking soda, salt and cinnamon.
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter and brown sugar until pale and creamy. Add the eggs one at a time, mixing after each, followed by the vanilla.
Add the flour mixture in two additions, alternating with the buttermilk, mixing until just combined. Fold in the diced peaches.
Pour batter into the prepared bundt cake pan and bake until golden and a toothpick inserted into the center comes out with a few dry crumbs, about 1 hour.
While the cake is still warm, use a thin skewer to poke holes into the cake, pressing the skewer all the way to the bottom of the pan. Space the holes about 1 inch apart. Generally brush the simple syrup over the top of the cake, letting it soak into the holes.
Allow the cake to cool completely in the pan, then turn out onto a plate. Serve with sliced peaches and more simple syrup, if desired.