I am not a lawyer.
But it looks to me like No, just running Bitcoin doesn't make you a "money transmitting business." To be a "business" you have to be charging people for your service.
If you're running Bitcoin to buy or sell goods or services in exchange for bitcoins, I'd say you're not in the money transmitting business.
However, if I were to start a company in the business of buying and selling Bitcoins, or that was a Bitcoin payment processing intermediary that took a percentage of transactions between buyers and sellers, I'd talk to a lawyer and jump through all the legal hoops (looks like here in Massachusetts I'd need a license and would have to post a $50,000 bond).
Or to put it in more concrete terms: I am not a money transmitting business when I use my credit card to pay for something from Amazon.com.
And Amazon.com is not a money transmitting business just because they accept payments.
However, Amazon Payments, Inc. (Amazon's Paypal competitor) is licensed as a money service business in a bunch of US states.