AmEx doesn't participate in dynamic currency conversion. Normally when you pay with AmEx (at least the USA varieties), you sign for the charge in local currency. Then AmEx converts the amount to your own currency (based on interbank rates) and you pay the bill in your own currency.

With a USA based card, if I book a hotel room in London, it's really priced in British pounds (not USA dollars for Americans) and my contract is to pay the amount stated in pounds. The hotel bills me in pounds, charges pounds to my credit card, and it itself paid by AmEx in pounds. AmEx itself converts the amount in pounds to USA dollars and I pay my bill in USA dollars, from my USA checking account.

When I book the hotel in London, I set the (usually SPG or Hyatt) website to show me the prices in British pounds and print the confirmation in pounds (and expect my email confirmation to also show the price in pounds). If I phone to make a reservation, for example through FHR, I always insist that the agent tell me prices in local currency (British pounds in this example), not dollars since the binding contract will be in dollars. Also, I never book prepaid rates or go through OLTAs, priceline, airline website offers for hotels, package tours, etc.

ADDED: A couple countries are exceptions. When I last visited Russia, they maintained a separate "currency" (and defined official exchange rates) for foreigners which international chain hotels (and some stores specializing in imported luxury items) used. International hotels in Argentina tend to contractually define their prices in dollars for foreigners and then convert to local currency at the time of checkout. [Argentina also has a recent law saying that certain travel expensive such such as tours and plane tickets must be paid in hard currency.] Turkey seems to define prices in dollars in some places but in Euro in other establishments. There was a time when inflation in Israel was so bad that many shops had prices marked in dollars and converted to local currency at the time of purchase, using that morning's exchange rate, but one paid in local currency (whether in cash or by credit card) unless one could negotiate a deal for paying cash in foreign currency. There are also cases near borders or with many tourists where shops openly advertise that they are willing to accept foreign cash, usually at exchange rates they post (for example, US$1 = CAN$1 at times for convenience if you drive across the border and don't bother too change money), but other countries (Russia officially, East Germany in the past) forbid locals to transact in foreign currency. Some countries forbid their citizens/residents/businesses to hold foreign currency or maintain bank accounts in foreign currency without special permission. Obviously enforcement varies.