I'm not sure when you will be in these locations in September, but it may not be as warm or as dry as you expect based on the footwear you plan to take. Back in the dark ages when I backpacked through Europe, I wore and took shoes that were up to the task of all the terrain and walking/hiking issues others have already listed as well as handling the balance and support requirements of carrying the pack on my back. I started my trip in September and never regretted having them.
I recommend researching the weather averages for the locations you plan to be walking around, keeping in mind that the times you are there, the weather could be better or worse than the averages. If you are prepared with a couple of good options, you can adjust at the time you are packing based on actual weather predictions for those places.
I'm not sure what you mean by "athletic" shoes, but I'm not big on things that look like running or exercise shoes either and I hate white soles on anything. I find hiking shoes with good support and some special insoles much better. There are many that are lightweight, breathable and water resistant or waterproof. Paired with the right kind of socks, they can keep you comfortably on your feet all day. You might look at this Gore-Tex website for some options http://www.gore-tex.com/products/women/footwear/. Most are athletic looking, but not all. REI and other stores that cater to outdoor enthusiasts also have good, often lightweight, selections for this purpose.
I'm also one of the 3-pair crowd and flip-flops are my 3rd pair for all the reasons others have stated. For a trip to Greece, and after trying them on in a different color at REI, I recently bought some of these Ahnu
Montara hiking boots http://www.ahnu.com/women-performance/montara-boot-nubuck/AF2128.html?dwvarAF2128color=ASAU#start=25&cgid=women-view-all. They are a dream on my feet, very supportive and I feel very stable in them. I have worn them here in hot AZ with socks and they are even comfortable in this hot, dry climate.
For the evening or after a long day walking around somewhat challenging terrain, I also got these Ahnu Karma flats http://www.ahnu.com/women-yoga/karma/AF2287.html?dwvarAF2287color=RDCL#start=4&cgid=women-yoga in a dark blue leather. Whether I take them on some more enclosed flats will depend on weather predictions for the places I'm going when it gets closer to the time we leave. I have some great sandals but chose these for their lack of toe-stubbing possibilities.
Both of these shoes get good reviews on Amazon, Zappos, Shoes.com, REI, i.e. all the usual online suspects.
Finally, you are likely to be amazed at the number of women wearing fashionable boots in all the locations you are going. On our trips to Europe, boots on women seem to be season-less. Boots I wouldn't think of wearing in the heat down here are commonplace in some of the hot locations we have visited.