From Heathrow? Hmm...

Tricky - there's no way to avoid going through central London, so buses and cabs may be a little to unpredictable assuming you're making a connection.

Personally, I'd do this:

- Heathrow Express to Paddington
- Bakerloo Line to Baker Street
- Change to Jubilee Line (easy change - walk across a platform)
- Jubilee Line to Canning Town
- Shuttle Bus (or no. 69 bus) from Canning Town

Other options include a taxi all the way (I'd guess at around £50 for pre-booked, a lot more for a black cab), or a taxi from Paddington to LCY (which leaves you open to the joys of the taxi queue at Paddington).

If you really want a low-cost option, you can use the Piccadilly Line instead of the Heathrow Express, and just change to the Jubilee Line at Green Park. This interchange is a bit of a schlepp, though.

If you have lots of luggage, well, get a pre-booked cab all the way. And leave plenty time (ask the cab company, they're usually pretty good at telling you how long these journeys take at that time of day). http://www.airportdirect.uk.com/ come recommended by many here, and are asking £56 for a pre-booked LHR-LCY journey.