The setup in your answer is a bad one.
First you parallel two batteries, which you shouldn't, because their voltages are never exactly the same. Their low internal resistance will cause a current from one battery to the other. So feed three LEDs from 1 battery, and the other three from the other.
Then you place all LEDs parallel, which means that you lose (12 V - 3.2 V) x 20 mA = 176 mW per LED in its series resistor, while the LED itself uses only 64 mW. Total power loss is more than 1 W. That's because the large voltage difference between battery and LED. The best way to get a longer endurance is to keep losses in the series resistors as low as possible. So better place two times three LEDs in series, so that their total current is 40 mA instead of 120 mA. The power loss in the series resistors is then (12 V - 3 x 3.2 V) x 20 mA = 45 mW per 3 LEDs, or 96 mW in total. That's less than 10 % of the power loss for all LEDs in parallel.
Then your batteries will last 100 mAh / 40 mA = 2.5 hours or 150 minutes. This is pretty optimal. The batteries' capacity is 1200 mWh, and the LEDs consume 384 mW, so with an ideal conversion you can get a little over 3 hours out of them. But the most efficient conversion using a switching current regulator will get you maybe 85 % efficiency, and then you only gain 9 extra minutes.
edit re comments
An alkaline battery's voltage quickly drops by 10-15 %, and then remains more constant for a great part of the discharge cycle. So either you calculate the resistors for a larger current at the start, and 20 mA for the rest, or for 20 mA at the start, and a lower current later on. The latter solution will give you a longer battery life, but a bit less brightness.
jippie suggests to use a switcher anyway to get more out of the batteries, and it's a thought. You'll have to place the batteries in series to get 24 V to allow a voltage drop as high as possible. The larger Vin/Vout ratio of the switcher will make it less efficient, but overall you should get some extra time from the batteries.
(1) Probable issue is attempt to massively over drive LEDs - see below.
Series LED resistors will be needed.
(2) Boost converter MAY be not working properly - see below for testing method.
LED datasheet here
TPS61201 boost converter datasheet here
Two x AA alkaline with provide a voltage between 3.2V and about 2V.
A larger value of C1 on Vin will do no harm and will help very low voltage/bad battery startup.
The TPS61201 boost converter will happily start and run on this voltage range.
The LEDs are NOT rated at 20 mA continuous - see data sheet.
LEDS are rated at
Also thermal limitations of IC must be observed.
Running LEDs directly off IC pins risks LED damage and possibly IC malfunction.
What is design requirement?:
Say 10 mA/LED and all on.
12 LEDS x 10 mA = 120 mA.
120 MA x 3V3 =~ 400 mW.
Efficiency at 120 mA out and 2V Vin ~= 75% - see fig4 from datasheet below.
So 400 mW/75% = 560 mW into converter.
At 2V Vin Iin = 560 mW/2V = 280 mA Iin.
This is well within IC capability.
So - IC is capable of providing requerement IF LEDs are correctly driven.
Problem may be excess LED drive OR bad converter components.
Test: Provide a 120 mA resistor load to converter.
R = V/I = 3.3 V / 120 mA = 27 Ohm.
Will converter supply 3V3 to 27 Ohm with 2V supply including startup?
Use lower R's for higher load current if desired.
If converter will not support desired load current then an inadequately rated inductor is the most likely problem.
Your Cout = 22 uF = 2 x data sheet value - should not be a problem. Sometimes high Cout can cause startup problems but 22 uF should be fine.
Most likely problem is massively high LED currents.
Add series resistors to set currents to 10 m max.
Note that Vf LED varies with colour - see datasheet.
ADDED:
New information:
LEDs are not as shown on diagram.
Assume max per LED current is 20 mA.
Actual LEDs are Dialight 5988710307F.
These LEDs are rated at 20 mA ABS MAX so you could run them at 20 mA, perhaps.
[Are you feeling lucky, punk?]
If so then double figures I supplied above to 1
20 mA x 12 = 240 mA.
This is still on the curve in Fig 4 above with 1.8V Vin so the converter can handle it.
TRY MY RESISTIVE LOAD TEST with R to suit real load.
If this passes OK then converter is OK.
If this fails then fix it first - chasing driver problems when the power supply is failing is liable to be unproductive :-).
ADDED:
A deleted answer suggested that rising battery impedance would mean that AA batteries could not be used in this application and that C or D cells were needed.
IMHO this is not true.
While AA cells will probably not reach full discharge potential due to falling current capability, they should work reasonably well.
The converter has a performance curve in fig 4 of the data sheet which shows operation at 1.8V = 0.9V/cell. As long as the batteries will provide the required load current at this voltage OR HIGHER the system will work OK.
At I_LED = 10 mA and all segments lit Ibattery at Vbattery=2V will be ABOUT 280 mA (see above) and at I_LED = 20 mA I_Battery at Vbattery = 2V will be about 550 mA (efficiency slightly HIGHER at higher load - see graph).
IF the battery is capable of providing about 500 mA at 2V+ then it will work. This is getting extreme for AA Alkaline but the battery will provide more power than this for much of its discharge life.
My simple resistor loading test will show whether the converter is OK at any given battery state.
Note that input capacitor matter muchly when batteries are near end of life. A large capacitor greatly reduces battery effective impedance on current peaks. Mean ESR is not altered but failures usually occur when current peaks occur during the boost cycle.
Best Answer
As noted in the comments, the best solution is likely to use 2 AA in series as opposed to 2 AAA.
Let's start with the knowns. Your MCU is likely drawing negligible current, unless you have all sorts of peripherals on that you don't need (timers, ADC, etc), or it is directly driving the LEDs (which is typically a bad idea, unless they are not pulling much current from the I/O pins).
Using 2 series alkaline batteries given you a voltage rail of ~3V. Alkaline batteries are 1.5V cells, but fresh ones will almost always give you a tenth or so more voltage, hence your measured 3.3V rail. This will not last all that long.
LED Stats
Your specified LEDs are rated for 3.2V at 20mA. Note, you do not need to push 20mA through these resistors if you don't need them to be that bright. This is a common mistake with simple LED designs. Reducing the current to 10mA per string will give you nearly the same brightness with double the run time. LED luminance does not typically have a linear relation to current draw.
On the other hand, you could also pulse the LEDs from the MCU. Turn them on for 1ms at full current and off for 2 or 3 ms. You can play with this PWM duty cycle to see how long you can leave the LEDs off before you really notice the decrease in brightness. Pulsing LEDs like this will save current since they are only on a portion of the time, and they remain a bit cooler, increasing their lifespan and efficiency.
Of course, the only way to limit the current without using an actual current driver is with series resistors.
Are you using series resistors? You didn't mention it, and another common mistake in these simple LED circuits is to go cheap and forgo them. This almost always a bad idea, especially if they are being driven from MCU pins. If you are using resistors, you have even less voltage to drop across the LEDs, and there was barely enough to begin with.
Doubling the Bank
Putting 2 more series AAA batteries in parallel with the original 2 would theoretically double the capacity of the bank; however, this is also not a good idea. Batteries don't always play well in parallel, especially not cheap alkalines. Parallel batteries can work fine with charge and dissipation controllers, but you will always run the risk of unbalanced batteries trying to discharge into one another constantly.
This is not to say it won't work (I have seen plenty of store bought products doing this) it's just bad design, and it will not perform as well as you had intended.
Doubling the Voltage
Putting 4 AAA in series will double the source voltage. Using linear regulation, this gets you nowhere, but with a good switching regulator, this can increase run time. Even still, it is not as good as just increasing the battery capacity to 2 AA.
One thing you could do with an increased voltage rail is put a very cheap LDO regulator to drop the voltage for the MCU power supply, and power strings of 2 series LEDs with series resistor directly from the 6.4V rail. To control them, you can sink the current into the MCU pins (active low) rather than sourcing it from the pins. This will halve your current consumption and increase run time; however, you have doubled the number of batteries - a net gain of zero.
Again, if you can afford the excess size of adding 2 more AAA batteries, you should just use 2AA batteries from the start.