I've been quite happy with the zero-fee savings account from Bhutan National Bank. The two BNB ATMs in Thimphu spit out the right number of crispy notes, the customer service is friendly and efficient, and their debit card seems to work at plenty of shops around town (with the exception of Tashi Supermarket - where the machine never works).
I also like their SMS service, which sends you a message automatically if a large transaction is made in your account (currently "large" means more than Nu. 5,000). Using this service, there are two ways to find your balance: you can either text the message BALA XXXX (where XXXX is your 4-digit PIN) to get the summary balance of all your accounts, or if you need the balance of just one specific account, you can use the message BAL ZZZZZZZZZZZZ XXXX (where ZZZZZZZZZZZZ is your account number, and XXXX is your 4-digit PIN).
Here's something I found out recently: Due to the intricate ways of the BNB computers, the response to the former message (summary of all accounts) takes a few minutes (even if you have just one account) while the response to the latter message takes only a few seconds. Thus, if you're in a hurry - use the second format.