Place the chicken breasts in a container with a lid or plastic bag. Add the next 4 ingredients as a marinade. turn the chicken to coat, and refrigerate for several hours, or overnight.
Remove the chicken from marinade, and grill until no longer pink inside. Discard marinade.
While the chicken is cooking, combine the cucumber, tomato, feta, and parsley in a medium sized bowl. Make a dressing from the last 5 ingredients, and add a little of it to the vegetable mixture. Toss and taste, adding more dressing until it is to your liking.
Serve the cucumber-feta mixture over salad greens if desired, and top with sliced, grilled chicken.
Notes
There will be some of the lemon dressing left over - store it in the refrigerator, and use it to top other salads or as a marinade.