New York (JFK) to Hong Kong over the North Pole (CX830/831). Then you take a short Dragonair flight to Dhaka.
Continental flies to Hong Kong non-stop using the same 16-hour polar route, flying from Newark Liberty (New Jersey). Again - the connection is by Dragonair.
Connecting from North America (West Coast):
Cathay Pacific, Thai & Singapore Airlines now have non-stop flights from Hong Kong, Bangkok & Singapore (their respective hub cities) to Los Angeles. Singapore Airlines may also have a direct flight to San Francisco as well. These airlines all have direct connecting flights to Dhaka from their respective hubs.
Connecting from Australia or South Africa:
You're better off connecting via the Bangkok or Singapore hubs (which are served by almost every airline -- it seems).
Connecting from Western Europe:
British Airways no longer serves Dhaka non-stop from London. Air India now serves Dhaka-London via a short stopover in nearby Kolkata.
Connecting from the region (Indian subcontinent or China):
Connecting through Indian hubs (Delhi or Mumbai) to Dhaka, because of the proliferation of Indian airlines, can involve delays - though as of early 2008 these are not significant. A large number of Indian airlines (e.g. Jet Airways, through several code-share agreements) have direct flights to Europe, and provide convenient connections to Dhaka.
Some regional flights like those operated by Thai Airways stop in Chittagong or Sylhet en route to/from Dhaka.
Nearby regional destinations like Kathmandu (Nepal), Paro (Bhutan), Kunming (China) and all Indian cities are readily accessible from Dhaka in under three hours and are served by a great number of private airlines. The most exotic destination from Dhaka is to Paro (Bhutan's Capital) and is served by Druk-Air, the national Bhutanese airline on Sundays (9:00 AM flight taking an hour). The approach to Paro Airport