Powering LEDs from Speaker Cable

No...
look up musicolour circuit.

 

mikeclegg, As Bluejets mentions. It's not 'currently' possible... at least not without some serious messing around.

As it stands, the signal down the line to the speakers is AC, and adding additional circuitry to it to rectify it, to store it, and use it for the LEDs would decrease the amount of power actually arriving at the speaker and will most likely create distortion of the original audio signal... meaning you get quieter, and crappier sounding music...
You have two options:
The Ideal Option:
Run a separate pair of wires to each speaker for power that you can use to power the LED's and the detection circuit to trigger them. Alternatively, you could use a battery. (Chargeable or disposable, it's up to you)
It's simple to do, and will require the least amount of electrical know-how and effort. Downside is it requires extra external wires, or batteries.

The Less Than Ideal Experimental Option:
Pay close attention to how I labelled this one!
Borrowing some concepts used for 'Power over Ethernet Adaptors'. You could re-route the speaker out from the stereo to a power injector which would create a DC-offset on the speaker wire. This speaker wire would now carry a DC voltage AND an AC voltage. You would then need to create a matching adaptor for each speaker that will then split the DC and AC voltages. AC will be sent to the speaker, and DC will be routed to the LED circuit you have constructed.
*Note: This could be potentially harmful to your speakers! If the DC voltage is not removed, it will induce a constant current in your speaker which will move it's 'resting' position, and cause it to heat up! This could result in physical damage from attempting to move too far, or electrical damage from the Voice Coils getting too hot and melting the insulation shorting out!... All very bad stuff... A matching filter to separate the DC voltage is a MUST.
The advantage here is that your can remotely power the speakers, and use a single pair of wires for both music and power.
The draw back is added complexity, and some distortion most likely to present itself in the lower bass notes depending on how the filter has been constructed. (Remember that filter components wont only filter DC per se... they will even filter slower AC, which would be lower bass notes. You can construct the filter to only begin doing this is a lower note than the speaker can play, but care needs to be taken to calculate these values.)

The experimental option will require some research, and experimentation, whereas the Ideal option will be much easier to get support and to find relevant documentation online.
Hope This helps!