If one has Atrial Fibrillation, what is the recommended medical procedure treatment?

Is there a medical procedure that mitigates the use of blood thinners?

Only if it is new onset. IF it is new onset (less than 48 hours), then you can have either chemical or electrical conversion. But rate control is quite important. One can live with A-fib a long as the rate is well controlled. If you have had it long term, you can go on anticoagulants for about 6 weeks. Then if they determine that you have no thrombi in your heart you can attempt electrical conversion. If there is a thrombus in your left heart, the risk will be too high for a piece to break free, causing a stroke. Hope this helps

If the patient is "in A-fib", especially if there's a rapid ventricular response, they will sometimes be given a drug to "cardiovert" them to a normal sinus rythm. If It's "paroxysmal" afib- or afib that comes and goes with no rhyme or reason- then chronic coumadin is often indicated.
In some circumstances, an Inferior Vena Cava filer, aka greenfield filter, will be inserted to prevent any clots that might develop from going back to the heart/ brain/ lungs. This is usually not done just for afib though.

Most Afib is managed with blood thinners and rate control.
There are two medical procedures that are being done now, although neither is a perfect solution. One is called a Maze procedure. It is done by a heart surgeon and requires open heart surgery. The other is called an Ablation. It is done by an Electrophysiologist and involves the use of radio frequency to "burn" and isolate areas of the heart which are thought to be the source of the Afib. Both are long and involved procedures, both carry their own set of risks, and both require hospital stays. Neither carries a 100% success rate.
Of the two, the Ablation is less invasive. Both of these procedures should be done by someone with plenty of experience.
Find more information at the Heart Rhythm Society webpage.

I had aFib problems... When I first had aFib, they tried to chemically cardiovert me with a large dose of Amiodarone. That didn't work, so I was electrically cardioverted. That worked for a while, but I would keep going back in. After a back and forth thing with that, I ended up getting the Maze procedure in September 2005. I've been out of aFib since.