And this is a problem because.....? You have to place a certain amount of faith in the manufacturing process and quality control from the factory.
Well, every EE professor I've had and every circuit theory both ideal and realistic I've seen say don't do it, but I looked up some LED datasheets at Fairchild and it appears that some (not all) LEDs with no resistors inside seem to have some ohmic qualities to them anyway, about 1.2v difference or so when you go from "off" to max spec current. That would make your balancing work.
So, for situations where you are sure you have LEDs that are all the same part number from the same manufacturer, have specifically chosen LEDs which have shallower I-V characteristic slopes and are almost ohmic devices, and have calculated that given their characteristics they will work in this configuration, sure go ahead. But that's not most of the people here.
Will you burn out more than one LED if you have an out-of-spec component in a group of 3? Maybe, but that depends on just how far out of spec that one actually is.
The tolerance for some of those LEDs is paper thin when they're powered by a voltage source instead of a current source, as the original poster found out. If the voltage over some of the LEDs at Fairchild changes by 0.1v and current is available, the LED will draw 20mA more! In a bank of 3, if one LED in the bank dropped 2v at 30mA instead of 2.06, it would reduce the current through the other two to 25 mA, giving that one LED 40 mA to deal with. That's a 0.06 volt tolerance for error. Whereas with a resistor per LED, if one LED drops anywhere between 1.7v and 3v instead of 2.06, that LED would draw between 20 and 33 mA and operate just fine, and any other LEDs would be completely uneffected by what happens to that one. That's a huge safety step compared to 0.06v.
Furthermore, if you try putting a green superbright LED from Fairchild in parallel with a red superbright LED from Fairchild, and calculate the resistor necessary to run both at 30 mA assuming they'll drop about 2 to 2.5 volts, you will find that a green superbright LED actually draws 2 mA at the voltage where the red one draws 30. The green LED would be off, and the red one would run at 58mA and burn out. And this is different colors from the same manufacturer, same series (Super Red superbright LED vs. Super Green superbright LED). With each LED on a separate resistor, even if the resistors were the same ohmage both LEDs would work even if you neglected to look up their voltages. Considering a lot of people here are guessing at the voltage their LEDs are based on whether they're white or colors and superbright or not, because they don't have any datasheets, none of those lights had better end up in parallel. I once burned out an expensive jumbo-size LED by trying to put it in parallel with the same color, smaller package LEDs.
But it should take about 10 years for any of this to be an issue in practical application of a properly designed circuit with parts that aren't faulty to begin with.
This is the BYOAC forum, right?

If you are the type that likes to poke around circuits with metal objects while the power is on,
This is the BYOAC forum, right?

The resistor arrays you refer to are very handy and a good choice. But they aren't very cheap. You also can't get these from Radio Shack, so tack on a few bucks for shipping.
I'm speaking of these things: (see picture) I'm pretty sure they're not expensive, but not knowing exactly what they're called I couldn't check on Newark to see for sure. I'm pretty certain the part I pulled one off of would not have had anything expensive on it.
And if you really really want to save on wiring effort or price, just get lights that have the resistor inside.