Shurlee, I'd guess that 90% of the sites on the web never requested a link, but many have thousands of links. How? Typically, they offer information that people find useful and think their visitors will find useful.
I'd agree that organic link growth is a lot slower than a highly organized campaign, but if your site has lots of great content, or even better, great tools or downloads, you'll steadily accumulate links from all kinds of sites.
Now if you have a 5 page affiliate site with no content or some regurgitated pages, you probably won't get many links by just being out there.
I think the important time to seek out links is when a site is launched. Some good early links can jumpstart traffic, and as the site gets more visitors it will start to collect links on its own.