I don't think that either Jim or I had a "favorite" part, so there's not a very clear answer to your question. I thought the scenery throughout was above average, but not spectacular. One highlight of the trip was that the rural Turkish people were so gracious and generous, and that is true any place where you are away from the tourist centers. Any Lycian Way town that is right on the water is a tourist center. The folks in those tourist towns were friendly enough, but it's a business arrangement - they have moved to those towns in order to work in the tourist industry, very different from the folks who don't earn their living by servicing tourists.
I think the thing I liked most about the route is the diversity. I've always loved hiking in US wilderness areas, but I've also come to love walking through settled area. Lycian Way is a great mix of English/German ex-pat coastal towns, cancerous growth of tomatostan greenhouse valleys, isolated paths with great ocean views, tiny shops in remote villages, jovial shepherds with their goats in the hills. Some of the least "attractive" places we visited were also the most memorable because of the local people we met.
Our favorite parts will not be the same as your favorite parts, but I'll describe them here to give some flavor. My most memorable experiences are listed below. You can grab our kml file and load it into google earth in order to see where these places are located: http://dl.dropbox.com/u/15393086/BPL/2011.04-LycianWay.kmz
Image numbers refer to the show: http://amyl.smugmug.com/Backpacking/Europe/Turkey-Lycian-Way
Day #2. A small shop in Gey, where we drank tea and bought supplies. Memorable because I couldn't figure out how to use hand gestures to indicate that I wanted hard-boiled eggs, so I pulled out my iPhone with audio Turkish-English dictionary and played the Turkish audio -- we all had a great laugh over that piece of technology being deployed for the purpose of acquiring boiled eggs. The lady at the shop went and told her neighbors and everybody laughed; then she went home to boil eggs while we drank tea outside the shop. Image 23.
Campsite #2, a fantastic perch above the sea. Images 24-26
The long walk along the Roman aqueduct on day #4. I liked it a lot because the aqueduct is in such good shape, some stretches still in use, and you follow it for miles. Others might not care for it because it's on the edge of town and a bit brushy. Image 42.
The Roman siphonic aqueduct on day #5, a very cool piece of engineering work: Images 50-52.
Campsite #6: images 70-73
Lunch #10 at the plastic tables on the side of the road, because of the extraordinarily friendly young men who gave up their seats and their food so that we could have a nice meal. We thought we'd have to eat the stuff we could buy at the little kiosk, but instead had a very nice meal. I don't think I've ever had a half dozen young men on the side of the road be so nice to me in my life. Images 101-103.
The little shop in the rather unattractive town of Demre, while we shopped the man at the auto repair stall next door brought us chairs and tea and sat down to drink with us. Never in my life have I had an auto mechanic go out of his way to make sure I had a comfortable seat and a cup of tea.
Campsite #10, in the riverbed just north of Demre. Very nice site on the river gravel, and special because a half dozen young men from the adjacent greenhouses came out to figure out what the heck we were doing, then called a friend who spoke a little english, then they all spent an hour trying to give us oranges and tea and have us in for meals and didn't we want to stay in their homes, and oh how they love Obama and so on. And the man across the river who was harvesting sand with a shovel who who repeatedly gestured that we should come to his home for dinner and a bed. In the States I wouldn't even camp in a river bed in the middle of town, but we did so twice on this trip and had great experiences both times. Images 106-107
The route btw Camp10 (at sea level) and Camp11 (at top of ridge). And the camp 11 itself. And then down the ridge back to sea level. Images 108-127.
Adouin's Gulls, on the beach near Finike. We didn't think we'd see them, as they are local and uncommon. They were even color banded, although we couldn't record the info without a spotting scope. Image 131.
Lunch #13, our picnic lunch on the beach at a fancy resort hotel. Fun because our scrappy appearance and scrappy food attracted the attention of numerous fancy guests who were far too kind and generous. A family of English speaking Turkish nationals who now live in Germany (here on holiday) sent their kids back to the beach to deliver fancy resort lunch food to supplement our meager cheese and bread meal. And a French speaking Moroccan family kept us company for quite a while. Image 134.
The scenery along the coastal walk around the lighthouse. Starting at the east end of the long Finike Beach, and wrapping around to Camp 14. Beautiful scenery, and more heavily hiked than most other sections. images 135-151.
Camp #15. At the ruins of Phaselis, just behind the beach in a small park, probably crowded with tourists during the peak season, it was deserted when we were there. Five star campsite just a couple miles from tacky Tekirova. Images 171-173.
We did not take the alternative route that goes on the backside (west side) of Mt Olympus. We spoke with a solo French woman at the end of the trip who said that hiking to the top of the mountain, for sunrise, in the snow, was the best part of her whole hike.
With the exception of a night in Kas and a night in Kalkan, we camped every night. We love to just walk, and drop our packs at the end of the day wherever we end up. We're very comfortable with that sort of travel, and prefer to sleep in the tent rather than in pensions. If we had chosen to accept the invitations, we could have spent every rural night in somebody's home, or in a shepherd's camp, but that's not our style. If that sort of thing suits you, then just go with the flow and see what happens.
There are quite a few major ruin sites and I think that any stretch you walk will include at least one or two major ruins. None of them really stood out as more interesting than the others, in fact, we both sort of preferred the scattered small random ruins that are found throughout the route, like this old sarcophagus - image 93.
There are a few places you should probably skip if you have limited time.
Looking at our route in google earth, after leaving camp #9, the first few hours eastbound, near the coast, was very nice. But then the route north, uphill, to the road (marked Gurses in google earth) is not very interesting. And the route from Gurses down to Demre had some problems because the malignant growth of greenhouses has made a mess of the trails. Also, when we were there, Myra, the ruins at the edge of Demre, had a dozen hawkers - grandmothers with their kids and grandkids, who would not leave us alone - in our faces like flies; made me feel sorry for the tourists who visit Turkey and never leave the tourist sites. If you are walking in this part of the Lycian Way, you should probably cross the footbridge (image 100, Lat 36.23064, Lon 29.9394) and continue east to the SW edge of Demre, and then take a cab to the spot marked "Camp #10".
The 20 km stretch of beach east from Finike is probably not the best way to spend limited time. images 130-134.
Tekirova is the only truly tacky beach-town we passed through. Jim called it "Las Vegas on the Mediterranean". We enjoyed it because it was so over the top tacky, but probably not worth it if you have limited time. Images 164-170.
I think my best advice in choosing which parts to walk is to avoid limiting yourself the coastal stretches that appear in photos (and probably recommended by guide books) to be scenic. The closer you are to the coast and/or the more dramatic the natural scenery , the more tourists they see, and the more commercial the interactions become. If you walk the whole route, you flip back and forth between coastal tourist towns and interior rural areas, the coastal tourist towns were good sources of resupply, but you don't really need to travel all the way to Turkey to be in a tourist town.
As Hartley has said, getting around by taxi (probably not cheap) or by dolmus (very cheap) was easy. The few times we needed to get somewhere by bus (btw the three legs of our trip), we just wrote down the name of our destination and approached a random person on the street and said, with a bewildered look on our faces, "dolmus?" and pointed to the name of the town. We'd soon have a collection of people escorting us to the place where we could catch the right bus, and making sure we got on the right bus. The long distance buses were quiet, comfortable, and luxurious, with snack and beverage service better than you get on the airlines these days. We only hitch-hiked once, from the eastern end of the Lycian Way into Antalya. We got a ride from the first car that passed, but it was about an hour before that car showed up. After a ~20 minute ride the driver flagged down a dolmus that was going into Antalya and transfered us, ensuring that we would make it to our destination.
I suspect you'll have a nice trip, wherever you go. Can't wait to see your photos and trip report! Amy