I finally found out what the problem was. By using F12 in Chrome to open the developer window I saw that I had the error
No 'Access-Control-Allow-Origin' header is present on the requested resource. Origin 'null' is therefore not allowed access.
So it was a matter of CORS, Cross-Origin Resource Sharing.
The reason I encountered this problem was because I was loading the HTML page and Javascript files from my C: drive without using a local server and then I tried to load the JSON file from a remote server.
(I forgot to mention this in my original post.)
Seems like all new browsers prevent this call, while IE11 (and presumably also older versions) allows it.
When I uploaded the HTML and JS files to the server it worked as it should.
Note
For Chrome you can open the browser from the command prompt with the parameters shown below:
chrome.exe --user-data-dir="C:/Chrome dev session" --disable-web-security
This will enable you to try this kind of calls during the development phase.