I've seen one other mention of 301 redirects not being allowed. That's on this page where someone wrote in the comments "...cause the policy of Google Adwords doesn't allow 301 redirect for landing page..."

I think he's talking about redirecting using PHP's header function to specify a location. This results in a 302 redirect unless another status code is given.

I can kind of understand why Google and others might prefer sending 302 status codes instead of 301s.

302 means the resource is "Found". This is supposed to mean there's a temporary change in location, but a lot of webmasters have used it in a way that ignores the "temporary" part of it.

Now consider what 301 stands for:

So maybe Google (and others) think that instead of redirecting when a resource has moved permanently, it would make more sense for advertisers just updating their campaigns with the new address. If you're using 301's with your campaigns, you're essentially saying "My landing page has moved. All references to this landing page should use the new url, but I'm not going to update my campaign even though I'm the one who's saying (by the meaning of a 301 code) that the URL should change".

I'm just taking a guess here. All I can suggest is to send 302 status codes if you heard that's what the PPC networks prefer.