I am installing three small ceiling fans for three separate rooms in my grandma's house, the only problem is that they are all connected to one circuit. Each room has its own switch and never interferes with other rooms, unless the breaker is tripped. I am trying to avoid running wire from each switch through the wall and into the attic and then to the fan. My idea was to connect these wires together in a new junction box in the attic.
Will this work? Or will I have to run new home runs from the panel to each room? Is there a way I can avoid this?

What is the amperage required for each fan?

What is on the existing circuit now, other than the three fans.

What is the amperage of the existing circuit?

Fan #1- 0.55amps
Fan#2- 0.59amps
Fan#3- 0.43amps
The circuit is 120v and is on a 15 amp breaker

I'm not sure what your trying to achieve.

Do you just want to run one feed to the attic and branch to each fan and use just the fans pull chain? That's OK.

Do you want a wall mounted switch for each fan and feed from the attic? That's OK too.

Power to a switch location and a 3 wire feed to the fan allows a lot more options such as timers etc. and devices that require a neutral.

Why not just wire the fans in place of the old lights and not run any new wire?

Ceiling Fan Wiring, Ceiling Fan Wiring Technical Help, Do it Yourself (DIY) - Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-wiring.php)