Batteries, if only they were ideal.
When you have a power brick for charging on your computer this device has a regulator circuit. This means that it will hold its output voltage over a wide range of load conditions. Batteries are not a regulated source, they change their output voltage as a function of current out even without changing the charge of the battery.
Lets look how much a battery will have its voltage change just during one use.
Now, before you use this persons technique, they were measuring the batteries voltage out of circuit after a recovery time. This in my experience is much higher then the actual voltage the load will see. As a battery fails they are very aptly modeled by a nearly continuous voltage source with a series resistance and a small parallel resistance causing leakage. This still brings the point across, as you use your battery it is very very quickly going to change voltage. Measurement techniques are mostly not required for this, but if you would like to know more, please feel free to ask on Electronics.SE.
Why does it not work?
Now if your computer is expecting a regulated input voltage and it sees a good voltage, starts to pull current and then drop off, nothing in your computer is going to be happy. I know this is a pretty long answer for something that can be explained with, the batteries are not regulated, but it is important to understand why. For example, if you succeed in finding a DC/DC regulator that accepts your battery voltage range as an input and outputs what you need for your computer input, you are ready to go.
For DC/DC converters, if you pick a linear regulator, or something termed LDO, you must have higher input voltage then output voltage. There are other regulators out there, for example, something like this DC/DC converter would do the job. The issue being that you need a very small input range, not ideal for dealing with battery power. I can fine 8 million DC/DC converters, but finding one that matches exactly where you want to operate takes quite a bit of time, so I will leave you with one that works and allow you to find exactly what you are wanting for the number of batteries you want to use. For example, here is a pretty long list of options.