Why would we need to ground an AC source?
Electric circuits are always a complete path, which is the very definition of circuit. In a DC circuit, current flows from one pole to another in a constant, direct manner. In an AC circuit, current also flows from one pole to another, but it changes voltage and direction many times per second. (With mains power, this is 50 or 60 times per second, depending on where you live.)

It's not correct to say that with an AC source "current is not flowing anywhere but rather just oscillating back and forth." Current is flowing, but it is changing direction rapidly.

Let's take a moment and do an analogy - mind you, not a great analogy, but one I think that may help. Take a band saw for example. There is a metal blade with teeth that is essentially a ring (a band, which is where the tool gets its name) wrapped around two drums. When you turn on the saw, the blade moves in one direction, and you pass material into the blade to cut. You can think of the band as direct current, always moving in one direction, making a complete circuit.

If you think of a hand saw, where you oscillate the blade back and forth, the analogy falls apart, because there is not a complete circuit. In order for electric current to flow, there must be a complete circuit. It may be that this sort of idea is what has confused you with regard to alternating current. Because it is possible to move a saw blade by only acting on it from one side, you might assume a similar effect could be achieved with electric current.

Instead, imagine that the blade must always be continuous, thus we're back to the band saw. This time, let's say the tool moves the blade up and down (like a jig saw) and the teeth on the blade are modified to cut in both directions. You can still pass material into the blade to cut, but you're always cutting with the same section of blade, assuming that the movement in each direction is the same. Each time the blade moves up, it has to stop and reverse direction. The same is true with the downward cut. Think of these momentary stops as the point at which the AC voltage source is at 0 volts (the zero crossing).

Unlike a battery, an AC voltage source is always changing the voltage potential of the two poles, in an equal and opposite manner. When one pole is at a positive voltage, the other pole (with respect to the first) is at a negative voltage. As the voltage of one pole changes, the other is mirroring it. Current flows from one to the other, always.

Now about that ground thing... When you consider an AC source and a load, you basically have two connections. One pole to one side of a load, and the other pole to the other side of the load. Neither pole is really ground, because they're both just opposite sides of the AC source. You can call one of them ground, but realize that doing so is just a reference. If you decide that pole "B" is "ground" then you're essentially saying all voltage measurements should be with respect to that pole. If you were to measure any part of the circuit with a voltmeter, you would connect the black probe to the wire you've labeled as "ground" and all readings would be based on that as a reference.

In most household electrical circuits and appliances, ground is actually a third wire. The two poles I mentioned earlier are called "hot" and "neutral" (black and white wires in the US), and "ground" is the bare copper wire (or often green). The purpose for this ground is safety. A simple appliance might have a metal enclosure, which is connected to ground. Hot and neutral enter the appliance to power it. If something were to go wrong, like a wire came loose inside, if the chassis were not grounded, touching it could result in an electric shock. Having the chassis grounded instead diverts current back to Earth ground, usually in a very abrupt manner, such that the circuit breaker trips and power is cut off.

Mind you, this is a simplified explanation, but I hope it helps you understand the basic concept of AC and to differentiate between what constitutes the wires of the circuit and ground both as a concept and a physical connection.

For more in-depth information about AC, check out this article at allaboutcircuits.com.