The flickering LEDs cannot be used in series, nor directly driven by AC. Think of the processor in each LED as being a bit like a computer– when you remove power (whether that’s by switching it with AC or by putting it in series with an LED that blinks off its power), it reboots, and never gets to run its flickering program.
However, there are some good ways to get the basic result that you are after. For example, one flickering can be used in series with one or more non-flickering LEDs, and will cause all of the LEDs in that series-driven set to flicker together. With several flickering LEDs, each in series with several more LEDs, you can make a wide and striking effect.
You can read more about this see an example application, here: http://www.evilmadscientist.com/2012/flicker-leds/
As far as the AC goes, I would recommend that you build a basic AC to DC converter, such as the “Full-wave Rectifier with Smoothing Capacitor” described here: http://www.electronics-tutorials.ws/diode/diode_6.html
This circuit can be made as simply as four diodes plus a capacitor, or with an off-the-shelf bridge rectifier (e.g., Jameco Part number 178001, $0.29) and a capacitor (say Jameco part number 93761). 6.3 V AC will give you roughly 4.5 V DC with this circuit– enough voltage to power two LEDs in series, or one in series with a resistor.
You could also run flickery flame kits directly from that 4.5 V DC. If your machine’s output circuit was rated for 0.25 A, then you would have enough power to run three Flickery Flame kits off of that one output. (Figure ~80 mA each typical draw.)
One note is that our white Flickery Flame kits are designed to run from 4.5 V DC, but the red/yellow Flickery Flame kits are designed to run from 3 V. So, three off-the-shelf white kits should work fine, but if you were using the red/yellow kits, you would want to substitute in a different resistor value, perhaps 150 ohms or so.