We will be in Rome, Florence, and Venice May 20-31. I don't think our apartment in Rome or Venice has air-conditioning, only fans (our hostel in Florence has a/c). I've seen that the avg. temp. for May is 74 in Rome and Florence and 70 in Venice. Does this sound about right? I don't want to be too uncomfortable in the non-a/c apartments! I have, however, lived in South Carolina my entire life and the heat here in the summer is almost unbearable, up to the 100's sometimes. So hopefully I wont miss the a/c too much for 70ish degree weather!
You definitely won't need AC!! enjoy your time there at the best time of the year.
Like all weather - very unpredictable. Four years ago we were in Rome last week of May/first of June. For three days it was so cold that the side walk cafe didn't open and we were a little unprepared. Return during the exact same time the following year, a little better prepared, and it was in the 90s the whole time. On average you should be OK but don't be surprised it very hot or very cold.
Definitely be prepared for anything and don't bank on the "average" temperature. The average for northern Germany in July is in the mid-to-upper 70s but when I was there last year, it was in the mid-90s for two weeks. When your trip is a couple weeks away, start watching the daily weather in the cities you'll visit (you can do that on a site like weather.com) and you'll have a much better idea of how to pack and what to prepare for.