Combine lime juice, sugar and a scant 2/3 cup of water in a glass or plastic pitcher. Cover and refrigerate for at least 2 hours (but no longer than 24 hours).
Rub the rims of 4 martini glasses with a lime wedge, then dip them in a dish of coarse salt. In a shaker, combine the limeade, tequila, strawberry puree and cucumber juice. Add ice and shake 10 to 15 seconds, then pour into the prepared glasses.