Bring a large pot of heavily salted water to a boil. Cook the fusilli according to package directions, then drain and rinse with cold water until completely cool.
While the pasta cooks, make the dressing. In a large serving bowl, whisk together the olive oil, balsamic vinegar, Dijon mustard, oregano, garlic, salt, and pepper until emulsified.
Add the cooled pasta to the bowl and toss to coat in the dressing. Add the cucumber, cherry tomatoes, red onion, feta, and chickpeas. Toss everything together until evenly coated.
Serve immediately, or refrigerate until ready to eat. This simple summer pasta salad keeps well for up to 3 days.