The story of III will definitely make much more sense if you play the first two games. In regards to whether or not you should, I would say that it is worth it, if only because going back to the first two after playing the third may be jarring because of small improvements in gameplay.
I'd start with the first two games for the following reasons: weapon access, puzzle/platforming complexity, and difficulty ramp.
You don't get access to a larger arsenal of weapons in God of War 1, as you only mostly have access to Kratos' Blades of Chaos and the Blade of Artemis (which is practically useless in comparison), in addition to four magic spells.
God of War II expands on this, giving you, in addition to the blades, the Spear of Destiny and the Barbarian Hammer. While many people still prefer the blades, some people have done full Titan playthroughs with the other weapons for the challenge of it. They're definitely not as useless as the Blade of Artemis and can be viable options if you so choose to level them. Again, you'll have access to an array of magic spells.
God of War III changes it up a bit by linking the specific magic spells to specific weapons. For example, you have the Blades of Athena and the spell you can use with them equipped is Divine Reckoning. If you switch weapons to the Blades of Exile, then you have access to the Army of Sparta. You also have more relics in this game, which give you more bonus abilities. If you start with this game, you'll probably feel that the older ones fall short in what they can offer you in regards to gameplay variety.
While I wouldn't necessarily call the games "cerebral," the puzzle elements do increase in challenge as you go through the series. Getting to know the types of puzzles the games throw at you before you jump into III would probably help you in completing them.
The platforming experience also improves. There's a notoriously annoying part in the GoW1 where you need to traverse these rotating columns of spinning blades. They thankfully learned from that and haven't had something as controller-throwingly irritating as that in the sequels.
There's a tendency for difficulty creep to happen in series, and this is no exception. The hardest mode in God of War 1 is not as difficult or punishing as the Hard mode for God of War III, let alone III's Titan mode. If you're someone who likes to replay games on higher difficulties, you'll probably want to start out with the first games so you can ease into the control schemes and less punishing gameplay.
Finally, there's the obvious point of simple technology advancement. It's been 6 years since God of War 1 was released, and while the game is very well done and certainly doesn't look bad, it's definitely dated. Personally, I have a lot of difficulty going back to older games if I've already played their successors, primarily because of gameplay improvements and tweaks that developers have made in the years during sequels because they learned from mistakes in the original.
Don't forget there's also God of War: Chains of Olympus and God of War: Ghost of Sparta on the PSP, if you have one, and they've also been released as the God of War: Origins Collection for PS3. Chains of Olympus is a prequel to God of War 1,
while Ghost of Sparta takes place between the first and second games.