The idea of “muscle bound meatheads” is largely an exaggerated stereotype that has a basis more in comedic effect than in reality. Studies have shown that full range strength training can improve flexibility (1, 2, 3, 4). Additionally, top weightlifters, gymnasts, bodybuilders, and others regularly demonstrate advanced levels of flexibility while being exceedingly strong. As long as you continue using your current range of motion, you have nothing to worry about.
Strength training does not inherently create inflexibility when done properly. Only an imbalanced program (such as a bad push:pull ratio) and lifting through incomplete ranges of motion can end in inflexibility.