What do I need to know to design an LED lamp
First, using a resistor to drop from 120V to 3V is very, VERY inefficient. You will be burning off about 90% of your energy in heat. You need a transformer here, at the very least.
Just doing the math: If you need 3A/3.3V for light, then 3A/120V is 360 Watts of power, of which you're extracting 9.9W for the actual diodes, and thus burn about 350W in heat in the resistors.

Second, LEDs in parallel is unreliable, because the actual voltage drop of the LEDs will vary a little bit because of process variance, and thus some of the LEDs will draw a lot more current than others. This will result in the lower-voltage LEDs burning out, and the higher-voltage LEDs being dim. To fix this, you need one current limiting resistor per LED, rather than a single one for the gang.

Finally, what's the point of wiring half of them each way? Why not use a bridge rectifier?