Bring ingredients for the simple syrup up to a boil, whisking occasionally. Refrigerate.
Take the lime wedges for garnish and and cut in the middle just until the rind. Place the cut in the lime onto the edge of a glass so it sits on the glass. Run it around the rim once and then remove. Dip the rim of the glass in salt, fill with ice, and put the lime wedge back on the rim for garnish.
In a cocktail shaker, muddle together lime juice, 2 sprigs of thyme, and blackberries.
Add tequila, simple syrup, triple sec, and 1 cup ice. Shake vigorously.
Strain over the glasses of ice and then garnish with a sprig of thyme and a fresh blackberry.