Looking to stay in London in a flat (modest studio or one bedroom) for 30 days in a location that has good transportation options. What areas would you focus on? Can you find one for $3000 a month or is that dreaming? What rental companies would you recommend?

Here are several apartment rental websites:

www.english-manner.co.uk
http://aplacelikehomelondon.co.uk/
www.priceapts.co.uk/
www.paramountrentals.com
www.citadines.com
www.londonaccommodation.net
www.yourstay.com

And when inquiring, I would ask if they offer a discount for a 30 day stay.

You might like to try estate agents. Usually they deal in longer rental periods, but you never know if one of their clients' would be quite happy to supply you with an apartment for a month.

Primelocation.com and search with W1 (most central) postcode, and you'll find lets and agents - email/phone them and see how you get on.

M_Kingdom2, any truth to the rumor that you might start renting out a few rooms in the north wing of your London mansion?

If style doesn't matter but price does, consider Oxbridge Apartments in Kensington. They provide very basic flats within walking distance to Gloucester Road Underground; just off Cromwell Road for the buses; grocery shopping, laundry, et al nearby.
www.oxrg.demon.co/uk
We're staying the month of October in a one-bedroom for about $3000. A studio would be cheaper.

Thank you for the feedback.

vcl, that looks like a nice area.