I just spent 5 weeks in Australia, and I eventually just gave up entirely on my Amex. While, yes, I was able to use my Amex at some (although not all) hotels and some (not all) higher end restaurants and locations, for the other 80-95% of the trip I encountered nothing but this garbage:
This is just one demonstrative example. Almost every POS machine had a hastily taped "NO AMEX!!!" note attached.
I travel 80% of the year, mostly in SE Asia but also in Europe and Latin America - and I live in Canada. Normally I just use my Amex everywhere, and occasionally I run into one or two shops or restaurants that won't take my card. Not a problem. But Australia just seems to have a serious hatred for all Amex cards. By the second week of my 5 week trip I just switched to paying with cash.
That's the end of my rant. I know there are reasons behind it. I'm just amazed that those reasons exist in 2019, and that in my recent experience I can use my Amex more readily in a small village market in Vietnam than I can while shopping in downtown Melbourne and Sydney.