I followed this tutorial, http://www.wadewegner.com/2013/03/creating-anonymous-rest-apis-with-salesforce-com/ , to create a public rest service in a developer sandbox. If I run any of the following:

curl GET http://sitedomainname.cs1.force.com/services/apexrest/Widgets/
curl POST http://sitedomainname.cs1.force.com/services/apexrest/Widgets/
curl PUT http://sitedomainname.cs1.force.com/services/apexrest/Widgets/
curl DELETE http://sitedomainname.cs1.force.com/services/apexrest/Widgets/

I get an HTTP 301 response.

But strangely, if I run that URL through POSTMAN, any request calls my GET method. And if I run it through SOAP UI, it calls my POST and GET methods correctly.

Finally, if I call the same url, but HTTPS, I get a HTTP 503 error.

Here is my class:
 

@RestResource(urlMapping='/Widgets/*')
global class WidgetController {

    @HttpGet
    global static String getWidgets() {
        return 'GET';
    }

    @HttpPost 
    global static String createNewWidget() {
        return 'POST';
    }

    @HttpDelete
    global static String deleteWidgetById() {
        return 'DELETE';
    }

    @HttpPut
    global static String updateWidget() {
        return 'PUT';
    }
}

I should also add that all of these methods work correctly when authenticated through workbench