ATM limit Up to € 840 / 2 weeks?

Can you please provide a link to that information? If I'm not mistaken the limit of €60/day was for native Greeks only, with no limits put on tourists using any bank's ATM in Greece. That limit may not even be in effect any longer.

I'm wondering if there is a failure in communication regarding wire transfers.

The starting point is your understanding how your financial institution handles international wire transfers. As a for example, Bank of America's International Wire Transfer :

To send an international wire, sign in to Online Banking. Opens a
modal window and select the "Using their account number at another bank"
link from the "Send Money To Someone" section of the Transfers
navigation menu and follow the instructions. Before starting, make
sure you have all the correct recipient bank details layer.

If you send your wire in foreign currency, your recipient (in most
instances) will receive the funds sooner than if you sent the funds in
U.S. dollars. In addition, initiating your transfer in local currency
will allow you to lock in the exchange rate up front; you’ll see the
exact exchange rate when you initiate a transfer.

You will need:

Recipient bank details

In order to complete an international wire transfer online, you’ll
need the following information about the bank receiving the funds:

Recipient bank name, address and country

Recipient bank’s routing code and recipient’s account number

SWIFT Code, National ID or IBAN number of the bank where the receiving
account is located

Purpose for transfer

The funds should be "electronically" transferred from your account to the recipient's account. The funds would be available within the time established your recipient's financial institution's clearing rules. How they get the funds out of their bank may be a different issue. If the Greek banks still have cash withdrawal limits for Greek clients, that would be different than foreigner ATM withdrawal limits. ATM cash withdraw limits will be controlled by the lower of your financial institution's daily withdraw limit or the foreign ATM machine, and any restrictions the Greek government may have invoked.

The withdrawal limit of your own bank is the most important limit to worry about if you want to use cash. Check with them and see what it is and if it can be raised to cover your expected withdrawals. Most banks will be happy to do so. And make sure you tell them you are traveling and when so they don't freeze your account when foreign withdrawals start coming in.

Second, each bank or ATM may set a limit on how much you can withdraw at one time. This should be per transaction though and as long as your bank authorizes the transaction you should be able to get what you need up to the limit per transaction. You can do multiple transactions or go to multiple ATMs to get the amount you need if their limit is too low. I have not heard of limits per card or account and the only thing the networks check is to make sure you have not exceeded the withdrawal limit put in place by your bank, but we are dealing with Greece here and things may be different requiring you to find an ATM operated by a different bank to get the cash you need.

The remark made about it being difficult to get cash from the bank is probably still valid due to the recent issues in Greece. From what I have read, it appears the Greek government is trying to force more of the country's banking transactions through the electronic route for tax tracking purposes and thereby made it difficult for Greek citizens to withdraw cash in any significant amount.

Greek capital controls are probably evolving weekly. It does appear that the Greek Finance Minister is loosing controls to encourage transfers into the country:
"...new cash deposits made in Greek banks, either in Greece or from abroad, which can now be withdrawn without any restrictions." Your hotel management may not be aware of the July 2016 report, or the rules may have changed again.

http://greece.greekreporter.com/2016/07/23/greek-finance-minister-further-relaxes-capital-controls/

Greek Finance Minister Euclid Tsakalotos has further relaxed capital
controls with a decision published in the government gazette on
Friday. This was the 11th ministerial decision referring to capital
controls, in addition to two Acts of Legislative Content, issued since
July 2015.

The measures chiefly concern new cash deposits made in Greek banks,
either in Greece or from abroad, which can now be withdrawn without
any restrictions. Cash withdrawals are also made generally easier
since the limit is raised to 840 euros per two weeks, instead of 420
per week.

The new measures also allow early full or partial repayment of
outstanding loans from financial institutions, allow Erasmus students
to open deposit accounts, allows the withdrawal of up to 30 pct of
money transferred from bank accounts abroad to bank accounts in
Greece, allows pensioners living abroad to open bank accounts for the
payment of their pension in Greece, while Greek pension and social
insurance institutions are allowed, in some cases, to pay pensions and
benefits abroad. (source: ana-mpa)

We currently live in Greece and have withdrawn as much as 1000 euros at a time in order to pay bills. You shouldn't have a problem as long as your bank increases your withdrawal limit to allow for it. And, we used a Piraeus bank ATM for the withdrawal.

Are there no hotels on the islands that take credit cards? Cash only = likely not paying taxes and having zero traceability. What hotel owner wouldn't want that?

That would be my guess - the "hard to get cash out of bank" is just a line. I know when we had the B&B we loved cash customers! Just sayin...

Thanks to all. I see the ATMS are no problem for the tourists. And that cash is, of course, not traceable.

I would at least ask them for a receipt and see if they even pretend to allocate some of that total for taxes. In Italy, the local taxes are all in cash (even if the room is paid by credit card) but they are very adamant about giving receipts breaking out the base rate and the taxes.

Interesting discussion. We were in Greece in June and were able to get cash from the bank cash machines easily. The only time we had a problem was when the cash machine was not attached to an actual bank. These machines are not serviced regularly and sometimes run out of cash. We were able to withdraw as much as 500 euros at one time the limit set by our bank in Canada.

Larger hotels will take credit cards. They will ask for cash but if you press they will take credit cards. All hotels have a sign in the lobby stating if a receipt isn't provided then the customer doesn't have to pay. Sometimes they forget so do ask for the receipt.

We were in a remote village in Crete and asked if we could use credit card. The owner told us that all his suppliers and staff demand cash payments. With the cash withdrawal limits imposed on Greeks a credit card payment hampers his ability to operate. We heard that story from virtually every shop and taverna. We just paid cash from then on.

mtcarroll it isn't clear to me if the hotel is asking you to pay them cash when you are there or if they are asking for an advance payment. If it is an advance payment then you really have no choice but to use a bank transfer, credit card or Western Union to transfer the money. That solution is the problem your hotel is trying to avoid.

However if the hotel is saying just pay cash when you are in Greece then you will have absolutely no trouble getting cash from an ATM machine.

Greece still is a predominately cash driven society. I am pretty sure someday the Government will impose credit card only payments simply so they can get their hands on more tax revenue. Until then you should expect to be paying cash at small hotels and tavernas.