Place cucumber, tequila, lime juice, and simple syrup in the jar of a blender; blend until smooth. Strain through a fine mesh sieve into a pitcher. Refrigerate until chilled, about 1 hour.Advertisement
Moisten the rims of three 6-ounce martini glasses with lime wedge. Place salt in a shallow dish. Dip rims of glasses in salt to coat.
Fill a cocktail shaker 3/4 full with ice. Add 1 1/2 cups cucumber mixture. Cover and shake 10 to 15 seconds. Strain until prepared glasses. Garnish with cucumber slices. Repeat process with remaining ice and cucumber mixture.