luggage in car concerns

Ask the hotel where you will be staying to hold your luggage until you check in. Do this especially in Athens. We have done this many times.

That won't work for the first day in Athens, we'll be staying at a monastery and can't go there until evening.

Please add some of your arrival/departure details so we can make suggestions.
When I traveled through the Peloponnese (small towns, archaeological sites), I did not concern myself with this, but I would not take the risk in Athens or Thessaloniki--especially if you are talking about a van where luggage will be in view (not two cars with trunks, right?)

Book arrival the night before. Then wouldn’t the monastery hold your luggage on your arrival day? If not, perhaps change your reservation to a place that will,

We will be in athens on a Sunday and again on a Wednesday.

Two rental cars, hatchbacks, so yes, it will be somewhat visible.

Leaving luggage visible in Athens is not a good idea. You need to do something different.

Maybe I'll try to switch to cars with a trunk

No maybe! Choose a car where you can leave your possessions out of view. If the car they give you doesn’t have out of view storage, request a different car.
Do not leave anything visible in your rental car ever. You will asking for a break in and theft of your belongings. Replacing your things in a foreign country is a HUGE hassle. We have had to do this.

ok I just discovered lots of storage lockers. I guess since we don't have them in the states anymore I assumed no one had them. I think that may the best bet for us. Seems cheap enough, there is even one right in the airport.

It;s better to ask the hotel to keep the bags for you for a while, or have them at all times with you. propably nothing bad will happen, but you have to be careful. Bags and suitcases have valueable things, and someone might try to open them or steal them.
Be careful and have a nice day!

yes that would be ideal but as I noted, we are not staying at a hotel, it is a monastery. We cant impose on them by asking them to hold our bags while we spend the day in the city (plus it would mean a 1.5 hr round trip to do so).

I think lockers are our only choice.