How to reduce 12v to 3.6v
You might get away with connecting sets of 3 LED strings in series.  At 3.6 V each, that would drop a total of 10.8 V.  By measuring the current drawn by a string, you can calculate what resistance to add in series to drop the remaining 1.2 V.

Depending on what kind of electronics is actually in each string, they may be OK with connecting 3 in series to 12 V.  That would supply each with 4 V.  Or, that could seriously overcurrent the LEDs, causing them to blow out or have severely decreased lifetime.  Without a datasheet there is no way to tell from your description.

Another option is the 12 V supply with a couple of diodes in series instead of a carefully calculated resistance.  Each diode would drop about 700 mV, leaving 10.6 V for the three strings, or 3.53 V for each.  Again, without a datasheet for the LED strings, it's impossible to say what is acceptable.

You also need to look carefully at this "12 V" supply.  Is that 12V&plusmn;100mV, or something unregulated like a transformer and four diodes?  If the latter, and the LEDs need a tight voltage, then you can't use your 12 V supply directly anyway.