Returning from a trip to the East, I'd like to share great experience using Bhutan Telecom's mobile Internet infrastructure. Using the 3G modem purchased at BT, I was able to connect to the internet from every place where there was BT mobile network access. Trongsa, Bumthang, Mongar, Trashigang, and even in the guesthouse at Rangjung, north of Trashigang. In Thimphu the speed is fastest (3G technology), whereas outside of Thimphu it is slower (GPRS/EDGE technology). The latter enables reading email and surfing the web, but a bit too slow for Skype phone calls.
To get mobile Internet, you must purchase a modem (around Nu. 5000, available in 12 installments), and then get the data plan (starting at Nu. 149/month with the new rates). For further details see the Bhutan Telecom webpage. Note, however, that the new rates are not yet updated on that page and can be found here.