I am Italian and not African American, but my wife is 100% Afro-Puerto Rican (or Afro-Latin American), therefore black, so I have some "indirect" experience traveling in Italy with her for at least 25 times.
Some might look at you with suspicion if they think you are an African immigrant and possibly (God forbid!) from a Muslim country (Italians don't like Muslims, and that sentiment goes back centuries). That is true also with law enforcement. But as soon as they find out you are an American, you will be immediately be treated like any 'White American' and any suspicion will dissipate.
Carry your US passport with you and wear stuff that will make you look unmistakably from the USA (being loud in the streets will achieve that goal too. Americans are very loud compared to Europeans). Therefore if you are an African American who likes to wear African attire, I suggest you leave that at home and wear something that makes you look like an American basketball player or, if you are not that young anymore, wear conservative business casual clothes. If you look like Kobe Bryant you will treated as a hero in most of Italy and people will stop you for autographs (Kobe grew up in Italy and people love him. He's often a guest on Italian TV where he speaks Italian fluently).
My brother in law, who's even darker than my wife and a male, has lived in Spain for several years. He said that in Europe he was never treated with the same level of suspicion he was treated in the US and never had any problems. Whenever he was pulled over by the police on his motorbike, the minute they found out he was an American, they wouldn't even bother to check his driver's license or his immigration documents. They would wave him good bye without further question. If he had been from Africa he would not have enjoyed the same treatment. Moral of the story, in Europe the color of your passport cover counts more than the color of your skin.