No, they cannot run off of AC
If a diode is placed in with a LED it will work on AC. The diode is to prevent the reverse voltage causing damage to the LED.
It is possible to replace normal Christmas lights with LED's. Since the LED's have polarity they have to be put in series with cathode to anode an so on.
The other requirements are to count up the voltage drops of the LED's, calculate a resistor for current limiting, and add the diode to ensure the LED's only have a 1/2 cycle of voltage to light them. You can mix colors in series, but keep track of the forward voltage and leave room for the current limiting resistor.