Using a box grater held over a kitchen towel, grate ½ the cucumber. With the towel held over the sink or a bowl, squeeze as much excess water as possible from the cucumber.
Finely chop the remaining half of the cucumber.
In a medium-sized bowl, combine Greek yogurt, lemon juice, jalapeño, minced garlic, chili powder, dill, and mint.
Fold in grated cucumber and chopped cucumber and let tzatziki chill in fridge for at least an hour before serving.