There are two methods I know of. First, you can extract the values from the form and submit them as a separate Ajax.Request. Alamar posted some good instructions on this:
Use Form.serialize(true) to get a hash
with parameter names as keys.
You can do whatever you want to that
hash, includeing deleting inputs,
altering their values, filtering them
and so on.
Ajax.Updated then will accept that
modified hash as a parameter.
Another method is to create an iFrame which will take the response from your form request, using the "target" property of the form element. If you're working all in the same domain, you'll have access to the contents of that iFrame and can use the returned information in there to update the parent page with any messages you want to return. This approach is more convoluted, but necessary in some cases. For example, if you're working with file upload inputs, the serialized hash in the first approach won't work.