Add water, lemon juice, cucumber, jalapeno, and honey to a blender and blend until completely liquefied. Taste and add more jalapeno and/or honey, if desired. Blend again if necessary.
Pour liquid through a fine mesh strainer to remove any larger particles and transfer mixture to a large pitcher. Place in refrigerator to chill for at least 1 hour before serving.
To serve, pour over ice and garnish with slices of lemon, cucumber, and/or jalapeno. For a fancier presentation, thread chunks of cucumber and jalapeno onto sprigs of rosemary to garnish each glass.