I've tried all the relevant options I can find, but still all new incoming messages in my inbox appear to be read. I want them to be treated as unread (i.e., appear in blue in the message list) until I've read them.

I'm running KMail 4.8.2 on ArchLinux.

I've searched high and low on the Web, but can't find anything other than a few very old bug reports. Most people have the opposite problem.

EDIT: I'm not running any filters. My e-mail account (GMail) is set up as POP3, not IMAP.

Are you using a "Show only unread" emails custom folder?
If so, try to avoid using it.