Catch them and hold them ransom.
Seriously, I understand your situation. Having careless, irresponsible neighbors is hell on Earth. Since you've already spoken politely with them, I suggest learning about your local laws on this kind of problem. Contact animal control and make a complaint. Document with photos/video of the chickens and the damage they do. Then prepare to take legal action. There is no quick answer to this situation.
Getting a dog is no guarantee, and it will probably dig up your plants. Cats will poop in the bedding so they are out of the picture, too.
Having a picture perfect garden is incredibly hard work. Most people don't realize the dedication it takes to get plants to maturity because they have never tried such an endeavor. Bad enough battling insects only to have a flock of chickens decimate a plant that will take years to recover.
This is an excellent reminder to those who have chickens and other livestock: Keep your stock in your own yard. Yes, we love our animals, but we don't love yours.
-agree 100%. It's unfortunate that you are having to deal with irresponsible neighbors. It is not incumbent upon YOU to do anything!! -sounds as if you've gone above and beyond in an attempt to be nice and cordial about the situation. You should be free to enjoy your own property, flowers, gardens without ANY interference from another's birds.
We live in a neighborhood that permits hens and roos. We do have a noise ordinance which prohibits habitual barking, howling, or crowing. Due to the close proximity of neighbors, we bring our roos in (laundry room) every morning from 6AM - 8AM to allow the neighbors to enjoy quiet, sleep in, or whatever they choose without being disturbed. (also, consulted neighbors prior to keeping 2 roos) Now, if an issue evolves and we're informed that we're disturbing folks - the roos will go (and we love 'em)! -BUT, due to the fact that we live inside a city limits (suburban area) it is OUR responsibility to enjoy our birds - not everyone else's. (If we wish to hear crowing outdoors, then WE will adjust and move to the country.)
-love my chickens, but also respect the fact that there are certain responsibilities and courtesies that go with flock ownership, so to speak. I do hope that you find a solution to this terrible dilemma, and get to enjoy your gardening. Keeping a healthy flower and vegetable gardens does take a lot of effort. It is disheartening to have another's birds destroy it.
-if there is any legal avenue per ordinances, codes, etc., it would be beneficial to explore those. Although it sounds like a rural setting, there should be some type of trespassing ordinance/law that might be pertinent. -best of luck, and I hope that you get to enjoy your property in peace!