the 70-LED strings get by with a resistor only.  The
100 LED strings
 appear to use a diode.   Heaven forfend that a diode bridge be used
 for them--that could cost additional cents per string. 
Ouch!. When I was designing the white LED stroboscope, I looked at
various LED data sheets on the web. They all emphasised the peak revese
voltage (which was quite low -- a few volts), and said that the LED would
be damaged if this was exceeded. Needless to say putting a resistor in
series and applying 110V AC is going to exceed that revese voltage.
 There's some fellow on the web who's been making various line/mains-
 operated lights using only a resistor in series with a single LED.
 Weird--I would have at least paralleled the LED with another
 resistor.  I'll try to find his web page, if anyone's interested. 
The normal trick is to put a diode in inverse parallel. The diode only
needs to have a PIV equal to the _forward_ voltage of the LED (think
about it, either the diode of LED is forward biased on each half cycle),
so just aobut any diode would do.  I'd probably use a 1N4007 since it
would survive if the LED went open-circut, and anyway, I have them in
stock :-)
-tony