I have been told the weather at of end of June / begin of July can be super hot and muggy and that going 30 days earlier; end of May begin of June it would make a huge difference. Agree or disagree? We will be going for 8 days on the ground, 6 spread over 2 islands (tdb) and 2 in Athens on the way out.
End of May-beginning of june is much better. Temp is almost ideal; 60s at night, up to mid-80s by 3-4 pm, 72 or so at sunset.... in 2013 I did this ... my last 4 days (June 14 - 17), it suddenly got much hotter, mid 90s, if I hadn't been at a beach, it would have been quite uncomfortable, certainly I wouldn't have wanted to be atop Acropolis, or walking all over Athens.
You might want to check out the RS Athens and Greece tour and click on its weather shown on its list to see what you can expect for average weather. We took that tour last year and went in the spring to avoid the heat and the mass of tourists from Europe. Do you have Greece guide book yet? If so, great. If now I would suggest you get one as soon as possible as it certainly help us in our planning. The RS guide book for Greece can be ordered elsewhere on this web site. We certainly enjoyed Greece and I hope you will too. Happy travels.
Rick's Guidebook is excellent on Athens, and on certain major highlights of the Pelopponnese -- the same things covered in his popular tours. The only island he covers is Hydra. So if Steve wants to go to any of the Cyclades Islands, he'll need to consult other sources.