It turns out that I’m having the spam problem because Gmail is doing something right:

Some organizations, like SpamCop and Sorbs, keep track of email spam so email providers/ISPs can use it to prevent spam from being delivered to their users. Occasionally, Gmail is listed on SpamCop because we do not reveal the IP address of our users. We don’t reveal IP addresses as part of our ongoing effort to protect every user’s privacy. Unfortunately, this can cause delivery problems for Gmail messages because some email providers/ISPs reject all messages sent from services listed on SpamCop and Sorbs.

I didn’t realize that they did that and it makes me like Gmail all the more! I just wish I could use it now.