Slice 4 very thin slices from the cucumber, then peel, seed and coarsely chop the rest. Combine the lime juice, chopped cucumber, some Simple Syrup and 2 cups cold water in a blender and blend until smooth. Pour into a pitcher and stir in the lime zest, mint and dill. Cover and refrigerate until very cold, at least 2 hours. Keep the cucumber slices covered and chilled.
Put some ice, a shot of vodka and some of the lime mixture in a cocktail shaker and shake for 10 seconds. Strain into a martini glass and garnish with a cucumber slice, some mint and dill. Repeat to make 3 more martinis.
Bring the sugar and 1 cup water to a boil in a small saucepan over medium high heat. When the sugar is completely dissolved, remove the saucepan from the heat and cool the syrup to room temperature. Refrigerate, covered, for up to 1 week.