I was thinking a 7.1 receiver might fit the bill, but can you control the volume of the channels independently?
... or can I at least turn them off?
You need to understand one thing. The Main Zone is the only powered zone. Zones 2 & 3 are non-powered, (pre-amp), zones.
This means you can simply run speaker wires from your AVR main zone outputs to your speakers. All AVR controls, like volume, work with the main zone.
If you do this, "Connect the 3 pairs of speakers to the FR, FL, - SR, SL, - SBR, SBL", you are using only the main zone, so volume and other controls work. Unfortunately, there is not an easy way to select which pair of speakers come on, or to control the volume separately. When you turn it on, you'll get the same thing and the same volume in each of the 3 rooms.
Or, you can wire speakers to Zone 1 (Main), then use Zone 2 for the second room, and Zone 3 for the third room. NOW you can control them separately.
However, remember that Zone 2 and Zone 3 are likely non-powered. It means the output from your AVR to Zones 2 & 3 has NOT gone through the amplifier in the AVR, so the signal is not strong enough to drive speakers. It is a "pre-amp" signal, and must be amplified. That means a separate amp for Zone 2, and another for Zone 3. (You can get them pretty cheaply.) With separate amps, now you can control the volume in rooms 2 & 3 separately by using the volume on those amps. And you can select different sources in your AVR for the different zones, ex. TV in Zone 1, Radio in Zone 2, CD in Zone 3.