(image from www.nasoneroma.com)
There are over 2000 of these water fountains, called 'nasoni', dotted around Rome. The water is perfectly drinkable, and often lovely and cool (thanks to all those underground channels).
There are a few maps detailing where all these fountains are in Rome, some of which are incomplete. The best I've found so far are on the acea website (they are a water utility company), here. You can even download a map to take around with you on your travels. Let me know if you find a better one somewhere.
All you need is an empty water bottle, and you're good to go!