Place lemon peel, fresh mint, and cucumber slices, in a clean glass jar. Add the vodka. Seal the jar and let sit at room temperature for 3 days. Shake the bottle twice a day.
After 3 days, strain the vodka to remove the fruit and herbs, and store in the refrigerator.
Combine all the ingredients in a cocktail shaker filled with ice. Shake until cold. Strain into a martini glass. Garnish with cucumber slice.