The new rules also prohibit climbing on the landmarks, washing pets, or swimming in the water during the summer period.
Raggi's office said the ban aimed "to prevent the incidents are contrary to rules of urban decorum, and to ensure adequate protection of the historical, artistic and archaeological capital of Rome", RomaToday reported.
It is also forbidden to throw anything, including water or other liquids, into the fountains. The only exception is small change; according to tradition, throwing a coin into the Trevi Fountain will ensure a return trip to the Eternal City.
The ban will be in place until the end of October - and anyone caught breaking the rules will face fines of between €40 and €240.
The regulation applies to around 15 fountains, including those in some of the city's main squares (Piazza di Spagna, Piazza Navona, Piazza Barberini, and Piazza del Popolo) and others which are located in popular gathering spots on summer evenings, including those in Trastevere's Piazza Santa Maria and Piazza della Madonna dei Monti.
Historically, Rome's fountains were often used for bathing and washing clothes, but these activities don't go down well with modern Romans keen to protect their priceless heritage.
The Trevi fountain - which reopened in November 2015 after a multi-million-euro cleanup - is one of the most popular targets for badly behaved tourists. After its re-opening, police presence was stepped up at the site and fines for bathing were increased, but this hasn't stopped determined visitors - and some locals - from wading into its waters.
Florence has also cracked down on tourists who choose ancient landmarks for a picnic stop.
The Renaissance city's mayor announced at the end of May that authorities would hose down the steps of Florence's churches during lunchtime in a bid to deter picnickers.