I thought I would write a quick post about the current money situation in Myanmar as of June 2015 as before I left I found information on the Internet and in my guidebook to be confusing and often out-of-date.
I waffled on the decision to go to Myanmar until I met a Swedish girl in Vietnam who assured me that money there was not a problem; a lot of the sources you find say you have to travel with oodles of brand new American dollars which is absolutely not the case anymore. I did bring $350 in U.S. money (which I obtained at a Thai bank by converting Thai baht to dollars) but I found that I didn't need it; after changing $100 at my guesthouse in Mandalay at a so-so exchange rate, I realized I was better off making a huge withdrawal from a Myanmar bank where the fee to withdraw money was around 5,000 kyat; I lost about $5 changing a $100 bill to Myanmar kyat so it's a lot better to lose around $5 taking out around $450 (which I did, and which last me the duration of my stay).
There are ATMs everywhere touristy in Myanmar. There are a half dozen ATMs at the airport, and dozens more around Yangon, Mandalay, New Bagan and Nyuang Shwe (the main city at Inle Lake), even at some of the major temples, like Yangon's Shwedagon Paya pagoda! I spotted a couple in Kalaw, too. I didn't travel super far afield, but I am assuming anywhere moderately large and on the solid tourist trail probably has an ATM or two. Yes, if you are going into a remote village somewhere (like during my trek from Kalaw to Inle Lake), you will need kyat to pay for things and there won't be ATMs around to access money. But, if you are prepared (e.g., you leave Kalaw with a small amount of spending money), you will be fine. Things in Myanmar are super cheap, so at most you're looking at costs like 500 kyat to buy a bottle of water or 2,000 kyat for a large Myanmar beer. ($1 = 1,100 kyat).
And yes, I was able to withdraw large amounts of kyat (first around $180 at the airport, then the $450 later on) and had a good exchange rate.
Myanmar ATMs accept VISA and Mastercard/Maestro. I only have a Mastercard debit card and it worked fine -- I had better luck in Myanmar than Kyrgyzstan, where VISA dominates the scene! The first ATM at the airport rejected my card, but the next bank worked just fine, as did the ATM I withdrew money from in Kalaw.
Money changers can change other currencies besides US dollars. Not every currency, but major currencies in the region, such as the Singapore dollar and Thai baht can be exchanged, as well as major world currencies like the Euro and the British pound. I had some Singapore dollars leftover and found a money changer who did the conversion at a great rate at a bank near my hostel.
Bring cash if it makes you feel better/more secure, but it's better to take money out of an ATM. I got a better conversion rate at the ATM and my USAA account will reimburse the fee since it was listed separately from the withdrawal (which is even better than the banks in Malaysia and Thailand, which build the withdrawal fee into the total amount withdrawn, meaning a bank can't the fee as a separate entity). I never spent all my US money and since I lost money changing baht to dollars for the 'security' of having a back-up should my ATM card not work, I'm reluctant to spend this money anywhere now in Asia and lose money again converting it back to another currency. (I'll spend it back home where it's legal tender.)
You should pay for everything in kyat, anyway. The people of Myanmar do their daily transactions in kyat. I think historically there was a time when the government wanted foreigners to bring lots of crisp new $100 bills so they could buy weapons or something, but not a single person in Myanmar insisted on dollars over kyat -- kyat seems to be the preferred currency, unlike Cambodia where most locals want you to pay in dollars and will give you back change in Cambodian riel.
So, there you have it. You don't need to travel to Myanmar armed to the gills with U.S. dollars; bring some in case something goes awry with your ATM card, but rest assured there are ATMs galore and you should not find yourself in dire straits while you are there (provided you have an ATM card from a well-established bank, of course).
If you have more information about the availability of ATMs in Myanmar or any questions regarding money in Myanmar, please share or ask in the comments!