I worked for a team in the IBM Linux Technology Center which had a very simple code review policy: all changes get mailed to a mailing list in git-format-patch form, and every patch needed at least one "Acked-by:" before pushing it (or two for changes to stable branches). Worked beautifully, by making code review not just painless but exactly like code review in a public FOSS project.
Yes, a reviewer just hits "reply" and either adds an "Acked-by: Full Name <email@example.com>" or responds to individual bits of the quoted patch with feedback on what needs fixing. Exactly the process followed on lists like LKML.
The patch author then adds the "Acked-by" lines to their commit messages (right after their "Signed-off-by"), and pushes the patches.
What is the incentive to do that? You are essentially going to stick your neck out (there can't be a bug in this, it was approved by Fullname, it says so right here) with no benefit to yourself.
"Acked-by" (or these days "Reviewed-by") most definitely doesn't mean "this can't have a bug"; it just means a second person looked over it and didn't see anything obviously wrong with it. As for why I'd do it: because it makes the software better, why else? Code review frequently catches bugs, and if they don't get caught now they'll have to get tracked down and fixed later.
For public open source projects there are lots of reasons to review and send an acked-by/reviewed-by line, at least some of which I think would carry across into a company-internal equivalent:
* if you review other peoples' patches they're more likely to review yours, which means your code is more likely to get committed.
* pointing out a problem and saying 'fix this' or 'have you tested X?' is cheap for you because the patch rework and retesting effort is on the other person. If you don't review and let something dubious through in a bit of code you care about and then have to track down and submit a fix later that's much more effort.
* it builds your reputation with other developers when you spot problems or suggest sensible design improvements. In open source projects this means internet fame. In the company-internal context it probably means better feedback at your next performance review :-)