CAPT OF A B737 MADE TURN IN THE OPPOSITE DIRECTION INSTRUCTED BY ARTCC CTLR DURING CRUISE WHILE CIRCUMNAVIGATING AROUND TSTM BUILD-UPS.

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ON CLBOUT FROM MCO; ABOUT 50 MI N OF MCO; VICINITY ORMOND BEACH; LEVEL 4-5 TSTMS PROJECTED TO BE IN FLT PATH. FO ASKED FOR DEV E OF WX; AND ZJX (FREQ 134.0) COMPLIED; ISSUED DIRECT CECIL VOR WHEN CLR. AS WE APCHED THE STORM; IT BECAME EVIDENT THAT A DEV W OF THE WX WOULD BE SAFER; DUE TO A LARGE OVERHANG AND ANVIL FROM THE STORM HDG TO THE NE OF THE WX CELL; WHICH WAS NOT APPARENT TO US PRIOR TO INITIAL RADIO CALL TO ZJX FOR PERMISSION TO DEVIATE. ATTEMPTED NUMEROUS CALLS TO ZJX (134.0) TO GAIN CLRNC TO DEVIATE W -- AT LEAST 6-8 CALLS; NO RESPONSE. IT WAS EVIDENT THAT CTLR WAS WORKING AT LEAST 1 OTHER FREQ BESIDES OURS; AS WE COULD HEAR HIS XMISSIONS TO OTHER ACFT. I DECIDED THE SAFEST COURSE OF ACTION WAS TO DEVIATE W; DUE TO PROX TO STORM CELL. ZJX FINALLY RESPONDED; TOLD US TO TURN R IMMEDIATELY; WHICH I DECLINED SINCE THAT WOULD PUT US DIRECTLY INTO LEVEL 4-5 CELL. (NO TCASII TFC WAS NOTICED.) WE CLRED THE CELL ON THE W SIDE IN ABOUT 2-3 MINS BEFORE INITIATING A TURN TO THE R; AND ABLE TO PROCEED DIRECT CECIL VOR. THE CTLR HANDED US OFF TO THE NEXT FREQ WHILE I TURNED TO THE R; AND ATC ASKED IF THE NEXT CTLR WOULD BE ADVISED OF OUR EMER INTENT TO DEV. I RESPONDED THAT WE WERE IN THE TURN TO CECIL; AND CLRING THE WX. NO FURTHER DISCUSSION OF THE DEV WAS NEEDED WITH THE NEXT CTLR (STILL ZJX). MAJOR FACTORS: FLT CONDITIONS - IMC; NIGHT. ATC -- INABILITY TO COMMUNICATE -- CTR WORKING TOO MANY FREQS TO BE EFFECTIVE.