Close, Three phase systems actually are a carry over from the days of early induction motors. Three phases can easily be connected to create a rotating magnetic field. We don't just use one phase at a time on the grid. All three are active. Phases A and B and C are all 120 degrees out of phase with each other. (That has nothing to do with the 120volts you know of in your house- it just happens to be another 120 number)
AC has inductive properties. I.e. it can induce electrical currents in wires across a core. (Transformer) DC cannot do this. Before vacuum tubes and other tech AC was one of the few technologies that could be easily stepped up to hundreds of thousands of volts.
Voltage and current have an inverse relationship. So we can step up voltage and reduce the size of the conductor $$$$ other feature is reduced line losses (HVDC actually beats us here but this is another discussion)
Having AC power allows all kinds of cool things! Did you know that your common household 120v feed is simply a single phase being split into two 120v 'legs'. Both 120 degrees out of phase. So if you remember elementary school math the abs() of phase A at its peak of +120 and phase B at it's valley of -120 = 240.
It gets better! We can be cheap on wire with AC systems! I can wire a 'multiwire branch circuit'. With phases a and b that share a neutral. If the current on phase a is 20 amps and the current on phase B is 20 amps what is the return current on my neutral? ( Google Mike Holt Multiwire Branch, Enjoy )
Want more fun. Read about Neutral Earth Voltage and NFPA 70 requirements for agriculture and equipotential planes. :-)
AC is freaking magical. The grid is even more awesome!