Lithium Polymer batteries (or LiPo for short) are a special case of Lithium-Ion batteries. They are sold in a 'pouch' format (and they are not the first to use a format like that, for instance, in the polaroid cartridges there used to be a battery in a similar form factor (polapulse)).
The reason why LiPo batteries are not so constrained in shape as their siblings is that the electrolyte is a liquid. Solid electrolyte batteries are typically much more constrained in terms of shapes.
This is also why regular car batteries (lead-acid) can be block shaped, the acid is also a liquid.
The polapulse was an interesting battery, it only had to fire the camera 10 times but was quite high current. The packaging constraints of the cartridge dictated the battery geometry and so a flat pouch like battery was born.
LiPo batteries exist in all kinds of interesting shapes and sizes too.
EDIT: looks like I was firsted somewhere below, doh!
Li-ion, due to the liquid, needs a hard metal case, whereas li-poly can often get away with a foil pouch. The surface area per volume of a cylinder is less, as well as easier to make structurally sturdy, which means less weight is spent in the casing. Tesla cares more about weight than volume, so this is an acceptable tradeoff.
http://en.wikipedia.org/wiki/Lithium_polymer_battery
Enjoy the read, I certainly did and it turns out the situation is a lot more complex (also historically) than I remembered it, but my memory wasn't quite as faulty as you made me feel initially (fortunately!).
Battery geometry is mostly dictated by physics and economics, even though a flat plate battery would be denser the gains in density would be off-set by the increase in cost and in the end the wasted space doubles as a cooling channel so you might end up having to separate your flat plates by periodic air channels anyway.
The reason why flat plates are better in some applications (for instance, for starter batteries in cars) is because those batteries are optimized for very high current during very short bursts of time. 300 Amps @ 12V is 3.6KW out of a very compact little box. The power generated from that little package is extremely impressive (if you pull the distributor cap on your car (if you have an old car) you can actually move the car for a bit on the starter engine in first gear or reverse!) but because of the short duration the power absorbed by the battery is relatively small (current is absorbed by the internal resistance of the battery during charge and discharge leading to a warming up of the battery).
But it does not last for very long so heat removal is not usually a problem, and the charge currents are reasonably low, so again no big problem in removing the heat (low current, longer time).
With electric vehicles the situation is reversed, the discharge time is relatively long (hours) but the charge times are getting ever shorter which means packs have to be actively cooled to avoid overheating, and the easiest way to do that is to flow air through gaps in the cells.
Plus they are mass produced, cheap, and have a relatively high energy density.
The new factory might work on creating some new battery technologies though, not sure. Ideally yes, you want the battery as compact as possible and lots of round batteries aren't the most efficient for that, but when you factor in cost and the natural heat insulation, it might have been the best option at the time. If it works, roll with it and work on one of the numerous other problems they have to solve.