Strain yogurt mixture by pouring into a coffee filter or double layer of cheesecloth set over an empty container to catch liquid as it drains (about 1 cup of liquid will drain). Cover and refrigerate overnight.
Shred cucumber and place in a fine mesh strainer set over an empty container to catch liquid as it drains (a few tablespoons of liquid will drain). Cover and refrigerate overnight.
In a medium bowl, stir strained yogurt, drained cucumber and remaining ingredients (salt through garlic). Cover and refrigerate for an hour or more to meld flavors. Serve chilled.
Video
Notes
I use a Chemex to drain the yogurt because we have one on hand, but you could also use an inexpensive pour-over #4 filter cone like you might use for camping.