[Graham]

As for the original question, I think if you really want them to exchange links with you, you should:
1) Have a site that is worth linking to. I get so many emails from sites with hardly any content and a very generic look. Everytime I get one of these I instantly delete them. If someone has a quality site I will gladly exchange links with them. I understand that we are not all web designers here, but if you don't know how to design a site, then spend the $150 to have someone do it for you. I think this shows how serious you are about being an affiliate.
2) Be honest, don't write paragraphs and get on your knees. Just say something simple... "Hey, I saw your site and I like it a lot. If you are interested in swapping links with my site let me know. Here is a link to my site. Keep up the nice work". Personalize it, tell them what you like about the site, maybe offer some suggestions. I think this will get you more link exchanges.
I don't really think there is much to it other than that... just avoid the template emails, it shows absolutely no effort.