So I was in a "two dollar" store and I saw that they had various cheap LED lamps. Out of curiosity, I bought one, described by the packaging as "12V MR16 1W 18LED", for $3.99. It looks like a typical MR16 lamp, but yes it does have 18 leds.
I tried it first on 12 volts DC, and then on 12 volts AC, but there was not even the faintest glimmer. OK, so it could have just been doa, but as it had been my intention to pull the thing apart, I proceeded to do that anyway.
Besides the 18 leds, there are four other components - a 1k resistor, a 1M resistor, a 0.15 microfarad 400V capacitor, and a bridge rectifier. All 18 of the leds are connected in series across the output of the bridge rectifier. The first input to the bridge comes from one of the power input wires through the 1k resistor. The second input to the bridge comes from the other power input wire through the capacitor, which has the 1M resistor connected in parallel across it.
In short, there is no way this is going to work on 12 volts. My guess is that the guts of this thing belong in a GU10 lamp that operates directly from the mains. I have to wonder if they also ended up with GU10 lamps with the MR16 boards in them.