We actually booked a Disney excursion for our Athens stop (however we did look quite a bit into a private tour for Athens). The main reason we chose the Disney excursion for Athens is because all of the private tours were long all-day tours that really appealed to DH and me, but we thought it would be too long for our boys because we are doing a 10 hour private tour in Ephesus, which is the day after Athens. We didn't want two 10 hour days in a row. For us, we chose the longer day for Ephesus as that appealed more to our family's interests than the long day in Athens.
So, for Athens, we are doing the Acropolis and Sightseeing tour with Disney that is just 4.5 hours long. This really was a tough one, because there is so much history in Athens, and one could easily spend a full day there. But, like I said, we had to pace our ports and we are choosing to not do two LONG days in a row.
And, we know we can always go back one day.
If you are going to Ephesus, we booked our private tour with Melitour. Highly recommended and excellent reviews & really reasonably priced for all you get (air conditioned car, you own guide, etc):
http://www.melitour.com/Ephesus.php
Then, we booked a non-Disney tour for Rome. We went with: Can't Be Missed Tours:
http://cantbemissedtours.com/
They are highly recommended in the Rick Steve's Mediterranean Ports guide (and I highly trust Rick Steve's recommendations). It's actually a walking tour of Rome, which is what we wanted (and this appealed more to our kids than a car). Also, you can't beat the price. They have been so nice with all my communications, and I talked with the Rick Steve's store and they told me more about this company. It is a very small family owned company that is fairly new (which is why there aren't tons of reviews online yet), but very good, very trustworthy, and does great with kids. You take the train (which sounded fun to our boys) into Rome with your guide. And, they leave Rome at 3:30 PM (gives 3 hours as a buffer to get back to port, even though it only takes 90 minutes if all goes well), so there is no concern with missing the ship.
I did research (and email) several private tours for Rome with cars, and my top two (if we had booked a private car instead of a walking tour) were: Rudy's Touring Service:
http://www.rome-car-tours.com/ and Rome in Limo:
http://www.romeinlimo.com/
As for our other ports, we are doing a Disney excursion for Malta (we are doing the Historical Hunt excursion, which is basically like a scavenger hunt around all of the historical sites, so a great, fun and creative way for the kids to see the sites and make it into a game), and the other ports, we are doing them on our own.