I understand that the client may not want that, but it may need to be explained that redirects are probably a good idea.
I have a few questions:
1. Do the old domains have links pointing at them, or any type of domain authority, domain age, etc? Might be worth it to park the domains on top of your new one, and do a redirect so Google doesn't index them both.
2. Why does the client not want the old domains pointed to the new one? If you still own the old domain, you can install the 410 code on the page, and request that it gets removed from Google in webmaster tools.
A brief update on what a 410 is:
"The Web server (running the Web site) thinks that the URL requested by the client (e.g. your Web browser or our CheckUpDown robot) is no longer available from that system. This is not a 'never heard of it' response, but a 'does not live here any more' response."
"The 410 error is primarily intended to assist the task of Web maintenance by notifying the client system that the resource is intentionally unavailable and that the Web server wants remote links to the URL to be removed. Such an event is common for URLs which are effectively dead i.e. were deliberately time-limited or simply orphaned. The Web server has complete discretion as to how long it provides the 410 error before switching to another error such as 404. "
We had to use this on pages that kept showing up for a local contractor. Once they were submitted to Google, we saw them removed within a week.
"When to use a 410 gone – error code?
If you intend to remove a page or file from your website and you very deliberately want visitors and search engines to know that it is really gone, you should use the 410 gone – error code. If you do not, and rather just delete the page or file, the visitors to your site will get a 404 – not found error which means that the URL you requested has nothing there. This should really ONLY be used if you are sure your intention is to tell the world this file is no longer here and to tell the search engines to take it out of their index."
You can read more here: 410 explained