UPDATED December 2016
While in the country, you'll use a mix of US dollars and Myanmar kyat. Most hotels, domestic airfares train tickets, and entrance fees will be in USD, while everything else (food, souvenirs, uses etc) will be paid in kyat. While in 2013 and 2014 a lot of places would only accept USD, this has changed in 2015 as the value of the local currency (kyat) has dropped 25% and the government doesn't want the locals using USD.
Until very recently (Nov 2012) there were no ATM's in Myanmar due to the economic sanctions on the country. This meant that you would need to bring all of the money you expected to spend into the country, with virtually no way to get funds once you ran out. And that money needed to be in pristine, unmarked, fold-free American USD (typically $100 bills).
Some of that has changed with the types of currency accepted for exchange and the availability of the first ATMs. Even so, I still recommend you bring all the money you plan to use into the country in cash, and use your Visa / Mastercard as a backup.
What Kind of Currency Should You Bring to Myanmar (Burma)
Along with USD, it is easy to exchange both Euros and Singapore dollars at the Yangon (& Mandalay) airport and banks. If you're exchanging funds in the streets, jewelers, or black market, you're best off to have USD.
Larger bills ($50 or $100) are exchanged at slightly higher rates than smaller bills (example 12/2014: 1030kyat / $1 for $50/$100 bills, 1000kyat / $1 for $1/$5/$10/$20 bills).
Do not bother with travelers checks.
How To Get the Best USD
Although the Myanmar government has recently told the banks to accept more than just the most pristine of foreign currency, bills that are not perfect may (or, more accurately "almost certainly will...") still be rejected, or exchanged at a lower rate. Treat your bills like gold... crisp, paper-thin gold.
Before You Go
Your local bank at home will likely be able to get you the currency you need before you go, though it is a good idea to call them a couple of weeks in advance to let them know you need pristine bills. If they don't have what you need on hand, they can put in a special order for exactly how much you need which can take a few days to a couple of weeks to arrive at the bank.
Some banks may tell you that this is no longer the case, as my Canadian bank occasionally tells me. Thank them for the advice and insist that you have the absolute cleanest and crisp bills. Go through the bills with them.
If you happen to be in Asia already, the exchange counters at the airports are the best and least painful options. Bangkok's main Suvarnabhumi airport is an excellent place to get the best USD. Bangkok's Don Muang airport, which is the hub Air Asia flies out of, is also a good option. You may have to try a few exchange places to get all the funds you need, so arrive early.
Bangkok Suvarnabhumi airport also has an ATM that dispenses USD that is located past immigration in the departure area. It's purple, and situated beside an exchange counter.
If you're in Bangkok or Chiang Mai, try "Super Rich" to exchange, but arrive early or call in advance as they often run out later in the day.
To make it as painless as possible, ask the teller for "US dollars for Myanmar." They will immediately know that you need to best bills and flick through their supply to find them for you. Be sure to check them yourself, noting for folds, blotches and smear marks.
Where Should I Exchange Money in Myanmar (Burma)
Contrary to the more recent Lonely Planet, the best place to exchange is NOT the black market. The best places to exchange are at the airport (during regular hours) and the banks. They offer the best rates and the security that you'll get what you should be getting.
Every time I arrive in Yangon here is always a tourist or five at the exchange counters wanting to only exchange $10 just to pay a taxi, then exchange the rest on the black market in the city. Don't be this person.
Banks are now found throughout the country, and even smaller towns off the main tourist track like Pindaya have a bank you can exchange at. Anywhere close to a tourist spot (Yangon, Mandalay, Nyaungshwe (Inle Lake), or Bagan will have standalone places for exchange as well. Note that most banks are closed on weekends and holidays and banks are closed for 1-2 weeks during Thingyan (the water festival every April).
You can also exchange money at your guesthouse or local jewelry shops, though the rate will likely be poorer (30 kyat/dollar less is fair) than what you would get at the bank.
Do not exchange money on the streets. This is a great way to get scammed, particularly in Yangon where the shady money changers often hang out near Sule Pagoda or the main Bogyoke Market.
When you leave the country, you can exchange your kyat at the airport or banks before you take off. No worries there. Keep in mind that no where outside of Myanmar will be interested in exchanging your extra Kyat, make sure you get rid of your extra before you leave.
After years of sanctions began to be lifted in mid/late 2012, modern-ish international banking for travelers began in November with a partnership between CB Bank and Mastercard that allowed withdraws from a handful of special ATMs in the major centers. A couple of months later, even more banking options for foreigners arrived with partnerships between Visa, a handful of local banks, and more than 100 ATMs in the country making it even easier to withdraw cash. Now there are thousands. It's a huge change.
That said, I've heard (and experienced) mixed things about people trying to use their Visa / Mastercard to try and withdraw funds here. Although the success rate seems to be improving, I've spoken with more than a dozen people who have had difficulties with one or more of their cards being rejected. Think of the ATM as a backup plan, and certainly not the single source of funds for your travels here.
If you do make a withdraw, the fee is approximately $5.00 USD (5000kyat) (similar to most Thai ATMs) and will dispense the local currency (kyat). The withdraw limit per transaction is approximately $300 but this may differ from bank to bank. This is in addition to the exchange rate fee you'll pay.
A growing number of businesses will accept credit cards. It's certainly worth noting that if a shop or are does accept credit cards, expect them to add a 3%-8% fee on top to cover their fees (and then some...).
Banks in Myanmar (Burma) have partnerships with Visa and Mastercard (and Chinese "UnionPay"). That means no Discover, American Express and no other funky cards for the moment. As always, it's best to call your credit card company and bank in advance to let them know you'll be traveling, so the chance of your card being blocked is reduced. You can usually find the number on the back of the card.
I carry Visa cards from two Canadian banks (TD & RBC), and neither of my TD Visa cards will allow me to withdraw funds in Myanmar, while my backup card from RBC has fortunately worked (and saved the day). The folks at TD have been unable to explain why their cards won't work here, and say it's not a problem of the transaction being denied on their end.