I would say the process would be about the same as if you were taking out a mortgage only it wouldn't take as long. No mortgage means less paperwork. On the other hand, it might take longer because the whoever is handling the money side of the deal would want to have a time period to wait and make sure the "cash" is going to clear the bank. Unless you're buying from the owner, the person handling the sell is going to want a cashier's check or electronic transfer and that takes time.
Can I ask you if you use a broker service for that deal, so I can give you more specific information? If it is a direct sale seller-to-buyer, you need a jurist and a bank account. He would explain you everything in details.
I have had the good fortune of doing this in the past, but not in the US. I suspect however that you'll be in the same strong position of negotiating a good deal and an excellent price. As soon as a seller or a real estate agent knows that your ability to purchase does not rely on obtaining a loan, you'd be surprised how accommodating they become.