Back in my carefree days of playing Pokemon Sapphire Version, I didn't really care about natures (at the time I was unaware of their stat-boosting and stat-reducing effects). Concerning Nature, I actually caught Pokemon based on looks (i.e., Swampert looks "serious", so I traded for a "Serious" Mudkip; Treecko's PokeDex says it is calm, cool, and collected, so I traded for a "Calm" Treecko).
My Torchic was a Docile Nature on my first-ever Sapphire playthrough, and I thought it was really neat.
At one point, I actually thought Rayquaza's nature was coded to match the Nature of the version mascot, because my Rayquaza had a Mild Nature on my first attempt catching it. When I caught Kyogre on my first attempt, it had a Mild Nature as well.
When I learned about the "true nature" of Natures; they became more of a concern, particularly because I kept yelling at my "Serious" Swampert for its crappy Speed stat and realized that the Serious Nature didn't give it any useful benefit. The only Pokemon I've hatched according to Nature and specifically EV-trained is a Hasty Flygon that I've nicknamed Trump Card. Finally, I've been able to outspeed quite a few Pokemon in Emerald's Battle Tower and Earthquake, allowing me to get past the 50-Trainer streak (a feat I've only achieved twice). So yeah, the Hasty Nature has been very helpful in my experience.
In the near future, I anticipate running Adamant and Modest Natures on most of my Pokemon (physical and special attackers), making adjustments for certain Pokemon (bulky Water-types and such). EV-training will follow...
And does the fact that I currently have two boxes in Emerald, each with an Abra (for Synchronize field effect) and Ditto (for Everstone breeding), for every existing Nature say anything about the importance of Natures...? ._. (while obtaining Ditto of every Nature in Desert Underpass, I actually came across a shiny Whismur, to boot)