Removing them from Google doesn't solve the problem of people going to the old URL from bookmarks or links on other sites. It also doesn't transfer PageRank.
You should be careful about site migrations/restructuring. Always plan ways to redirect users looking for old URLs. A 404 is the least ideal option to show users. And even then, it should be a custom 404 that has information on how to find the resource that they were looking for.
There's really no point in deliberately removing a page from Google unless it's content that you don't want to be indexed. Otherwise, a simple 301 redirect will cause Google to update their index (remove the old link and add the new one) and transfer your PageRank as well. It's generally best to keep the 301 redirect up as long as possible (since you're not changing domains, you can keep it up indefinitely) so you don't lose as much traffic. And it also gives people time to update their bookmarks/links.
That said, a custom 404 that informs the user that the resource has been moved to a new location would be better to get people to update their bookmarks/links, since a 301 is pretty transparent, and most people may not notice the redirect.
Ideally there'd be a way to put up a "301 page" that tells both search engines/bots/browsers and people that the page has been moved to a new location. Because right now it's basically an either/or. You can either create a 301 that informs search engines/bots/browsers the page has been relocated, OR you can create a 404 that informs the user that the page has been relocated. But there's no easy way to do both.