Wire splitting to switches
Yes, you can do this. The 14/2, with a ground, would be run into your 3 gang box. The black hot would be connected to the three black dimmer switch wires using a wire nut or appropriate connector. Then the load wires from the switches would be hooked up to the appropriate black wires from the lights. The white wires, neutrals,from the lights would all be connected to the white wire from the feed. The ground wires from the lights would be connected to the ground wire from the feed and from the ground wires from the switches. Good luck.

Since the dimmers are separated, run a 14/2 with a ground to the second and third switch box. In the first switch box connect the black feed in with the black feed out with the dimmer hot, connect the white wires together, connect the light load to the dimmer and connect the grounds. Wire the second box the same way. The third box connect black feed to dimmer, white neutrals together, light load to dimmer and connect grounds.