Renee, Ahhh, if only it was that easy. I have a 1/2 dozen short sale listings in negotiations right now. I have gotten an approval in as little as 21 days, and I have waited almost 6 months on others!
A cash offer is treated like any other offer in the short sale process. The list agent is responsible for submitting that offer, along with a package of information about the seller (hardship explanation, financial proof, etc) to all lien holders. The lien holder(s) can take 30 - 90 days (on average) to process the short sale application. When the lien holder approves the seller's package, and the terms of your offer, then you begin a normal escrow period. With a cash offer, that can be 5 - 15 days, depending on the type of property and the contingencies in your offer.
Patience and consistent follow up (to the listing agent) are the key to a short sale -- and keep your eyes open for other possible properties, because there is no guarantee the lien holder will approve a short pay.
Best of luck! Grace