In a fine-mesh strainer over a medium bowl, add cucumber and 1/4 teaspoon salt. Strain cucumber 30 minutes. Hand squeeze remaining water from the cucumber (I use a paper towel for this).
In a large bowl, stir together cucumber, remaining 1/2 teaspoon salt, yogurt, garlic, vinegar, and dill.
Slowly stir in oil until thoroughly combined. Refrigerate until cold or serve at room temperature.