I'm always a little nervous about leaving my luggage in the trunk when I go off to explore, but I do it--only if there's an actual trunk or luggage cover.

I think I'd remove anything that would identify the car as a rental car.

Depending on your rental car company, it could be a problem as some of them love to advertise. If the car has a trunk, and you can back it up into a parking garage, then it lowers the risk, but we normally do not stop anywhere with luggage in the trunk unless we know the area, and it is not overrun with tourists. It's too easy to make yourself a target.

We have always had our stuff in the trunk of our car wherever we went for sightseing in Turkey and never had any problems. And that is the last thirty five years.

You should be more careful in large cities where you may be observed loading your car, followed and then robbed. However, even that is rather unusual in Turkey. Nothing like Spain and Italy where there is incredible risk.

Priene and Miletus are in the middle of nowhere (near small villages) where you will be 100% safe. Didyma is close to a touristic area and the only danger I can envision would be from some of the tourists.

In any case, I strongly recommend that you do not drive in to Didyma but do Herakleia/Kapikiri, on lake Bafa, just some miles off the higway to Bodrum from Miletus as you pass the town of Bafa, on your left.

I have never heard of a rental car seen as a target in Turkey.
However, valuables left on the seats of unlocked cars will attract petty thieves anywhere in the world.

I agree with OC. You shouldn't have any problems. I've visited about 100 archaeological sites in Turkey by motorbike and have always just left by helmet on the bike (and backpack and jacket a lot of times). Only once did I arrive back to find my helmet missing but the man from the village tea house came rushing out with it saying he'd been looking after it.

Obviously in cities I take greater precautions.

Although I didn't visit these particular sites, I did visit others. I took some basic precautions, like making sure that I didn't open the trunk once on site until ready to leave, and I didn't leave anything valuable in a place within the car that could be seen from outside.

As I had been told to expect in advance, there were a few places (not many) where I parked where a person watched the cars, and that person asked for a nominal fee at the time I parked. I was glad to have been alerted beforehand to this possibility, as I would otherwise have found it very unsettling to have someone come after me, insisting on money. It only happened in a few places, and in each there were lots of other people around, so I didn't feel as threatened as I might otherwise have felt. Perhaps otherchelebi or Croesus can confirm how common this practice is.