Thanks, Transient. As it turned out, deleting the two files in the System>Library>Keychain folder eliminated the keychain password prompt I was getting in Safari, but it was still there in IE.

So I did go to Applications>Utilities>Keychain, but the repair didn't work, because it prompted for the keychain password which of course I didn't know.
But choosing Window>Keychain First Aid, then Options, Reset my Keychain (back to factory) seems to have worked and I'm not getting the prompt in IE. If it comes back, I'll post a message. Thanks.