Choosing leds for batteries power
If the LEDs are to be directly powered by the batteries, the blue, white, and possibly the green LEDs (depending on which type of green LED it is) will not work at all on 3 Volts: They require a forward voltage of 3+ Volts, 3.4 Volts as specified in the question.

Also, once the voltage starts dropping due to depletion, even the marginally lit ones (e.g. the green, may be the orange) will no longer work.

Therefore ideally only red LEDs should be used: Red LEDs typically have the lowest operating voltage amongst visible colors. Alternatively, as noted, 3 cells would be needed in series.

Now about power:

LEDs can be operated typically at half the nominal current or less. So 20 LEDs operated in parallel at around 10 mA each will require 200 mA current, well within the capabilities of typical AA Alkaline cells. 

For instance, the Energizer E91 Alkaline AA battery is rated for around 3 to 4 hours at 200 mA current. Put three of those in series, add in an appropriate current limiting resistor for each LED, and it will all work.