B737-700 FLT CREW ENCOUNTERS WINDSHEAR AND DIVERTS TO ALTERNATE.

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DEPARTED FOR ZZZ WITH LANDING ALTERNATES OF ZZZ1/ZZZ2. PLANNED ARR FUEL OF 11.4 WITH 3.1 TO ALTERNATE. ARRIVED IN ZZZ TERMINAL AREA WITH ABOUT 11.7 AND LOTS OF CONVECTIVE/TURBULENT ACTIVITY IN THE AREA. A LINE OF TS STRETCHED WEST OF ZZZ. LONG VECTORS AROUND FOR ILS DUE TO WINDS. WE WERE BEHIND ANOTHER CARRIER ON FINAL. EVERYONE APPEARED TO BE GETTING IN. THE OTHER CARRIER WENT AROUND AT ABOUT 200-300 FT DUE TO WINDSHEAR WARNING. ON FINAL WE WERE EXPERIENCING VERY LARGE THROTTLE MOVEMENTS TO KEEP THE ACFT ON SPEED. WE NEVER RECEIVED ANY WINDSHEAR ALERTS; BUT I ELECTED TO GO AROUND AT ABOUT 500 FEET DUE TO APPARENT WINDSHEAR. WHILE BEING VECTORED AROUND FOR ANOTHER APCH; I CONTACTED DISPATCH ABOUT AN ALTERNATE. DISPATCH RECOMMENDED ZZZ3 SINCE THE MAIN WX WAS STILL TO OUR WEST. ZZZ3 WAS APPROXIMATELY 2000 OVC WITH STRONG WINDS. EVERYWHERE HAD STRONG WINDS. BURN WAS 3.1. DURING OUR VECTORS APCH SAID TWO B737'S LANDED AND A B757 WAS ON FINAL. HE WENT AROUND DUE TO A WINDSHEAR WARNING. WE RECEIVED DELAY VECTORS TO ALLOW THE WX TO CLEAR OUT SOME. NEARING OUR DIVERT FUEL OF 8.1 WE ATTEMPTED ANOTHER APCH BUT DID NOT BREAK OUT AT MINIMUMS. WE WENT MISSED WITH 8.1 AND IMMEDIATELY REQUESTED VECTORS FOR ZZZ3 (WE HAD PREVIOUSLY GIVEN THEM A HEADS UP THAT ZZZ3 WOULD BE OUR ALTERNATE). WE RECEIVED STANDARD ROUTING AND ALTITUDE (FL230) WITH ARRIVAL FUEL OF 4.5 (VICE 5.0) ONCE EVERYTHING WAS IN THE FMC. ZZZ3 WAS LANDING RUNWAY XY SO THE ROUTING WAS ACCURATE. WE PULLED UP THE METAR FOR ZZZ3 AND THE WX WAS AS PREDICTED. SHORTLY THEREAFTER (ABOUT HALFWAY TO ZZZ3) WE RECEIVED AN ACARS MESSAGE THAT ACFT WERE GOING MISSED IN ZZZ3 DUE TO WINDSHEAR. I CALLED DISPATCH ON THE RADIO TO FIND ANOTHER ALTERNATE. I RECOMMENDED TWO ARPTS; BUT HE SAID THE WINDS ALL THROUGH THE AREA ARE BAD. HE RECOMMENDED ZZZ4 (STRONG WINDS AND 2000 OVC - LIKE ZZZ3 - BUT AT LEAST IT WAS NOT IN THE SAME AREA) SO WE PUNCHED IT INTO THE FMC AND IT SHOWED ARRIVAL FUEL OF 3.0. WE DECLARED AN EMER DUE TO LOW FUEL AND REQUESTED IMMEDIATE CLB AND DIRECT ROUTING. THIS TIME WE GOT IT WITHOUT HESITATION FROM CTR. WE BLOCKED IN WITH 3.2 (TWO LOW FUEL LIGHTS). NOT IN DANGER OF FLAMEOUT; BUT WAY TOO LOW FOR MY COMFORT LEVEL.