It depends. This april it was warm, but may it's cool
For city visiting only rain would make it a challenge, otherwise it does not matter, really
I don't think its so easy to predict the weather between April and May and especially between those three cities. Weather is different between Venice/Florence/Rome :) If those are your 2 travel months, I'd pack a rain jacket no matter what and not worry about the weather! Will be great no matter what the weather does in the city.
As others have said, the weather is variable that time of year--from year to year and from place to place. We were in Umbria and Rome in May of 2015 and didn't see a drop of rain in 11 days of travel. Temps were warm to hot. We went back In May of 2016, again to Umbria and Rome. Umbria was cool and rainy the entire time we were there. Lots of locals were still wearing their winter "puffy" coats. Rome was warmer and sunnier. The weather in March of that year was warmer than May. Go figure.
So, toss a coin and be prepared for all sorts of weather. I do like May in that you'll have a bit more daylight, but you will still likely encounter a range of weather.
Since we don't know where you are based, it's even harder to advise as we don't know what you consider cold weather. As Katherine has pointed out, Italians may well be wearing their padded jackets when others would be in T-shirts and shorts!
holiday-weather.com gives you averages (temps, rainfall etc)