Are you using SharePoint Online? We encountered this same issue with SharePoint Online recently that it started returning 403 errors to requests that have the "Origin" header set, which is the case for Postman as well as a few similar Chrome apps such as "Advanced REST client" and "REST Console." Although it may be the case that it was the behavior of Chrome that changed and it only recently started adding the "Origin" header. In either event, the Origin header is what is causing the issue.
I haven't figured out a way to disable Chrome from sending the Origin header, although there may be a way, I didn't do too much research. My workaround has instead been to use SoapUI (http://www.soapui.org/) to do my SharePoint REST testing. Note that it isn't as friendly a solution, and if you're using SharePoint Online, you'll need to log in with your browser and then copy the cookies and add them to the headers being sent by SoapUI. But the advantage is that you're in full control of what headers SoapUI sends, unlike with Chrome.
Another option that I haven't tried myself would be to use a Firefox plugin instead. I read that Firefox doesn't send the Origin header in as many cases as Chrome does, and there do exist REST client plugins for Firefox. But again, I haven't tried that myself as SoapUI works for me.
Hope that helps.