A B737 ENCOUNTERED MODERATE CHOP AND RAIN LEADING TO A FLAMEOUT OF THE L ENG. THEY DECLARED AN EMER; DSNDED; AND RESTARTED THE ENG. THEY THEN DIVERTED TO NEAREST SUITABLE ARPT.

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WHILE AT CRUISE ALT AT FL390; WE ASKED FOR A SHORT CUT. ATC GRANTED OUR REQUEST WHICH KEPT A LARGE TSTM WELL R OF OUR DIRECT COURSE. NEARING YYY WE NOTED SOME SMALL LIGHT GREEN RETURNS ON OUR RADAR; SO WE ASKED ATC HOW THE RIDE WOULD BE. THEIR RESPONSE WAS 'ONLY LIGHT CHOP; NO PROB.' AS WE PROGRESSED; THE LEVEL OF INTENSITY BEGAN TO CHANGE; AND WE NOTED A FLASH OF LIGHTNING FROM BELOW US. THE FLT ATTENDANTS WERE DIRECTED TO TAKE THEIR SEATS IMMEDIATELY; AND WE ASKED FOR A CLB TO GET OUT OF SOME MODERATE CHOP. RAIN WAS NOW HITTING THE ACFT. THE IGNITER SWITCHES WERE PLACED TO 'FLT' FROM 'CONTINUOUS' (AS WE HAD BEEN IN THE CLOUDS SINCE DEP). THE FMC SHOWED OPTIMUM ALT OF FL407 WITH MAX ALT OF FL410. AS WE STARTED THE CLB; I NOTED THE L ENG HAD BEGUN TO ROLL BACK AND WAS FAILING. NOT ABLE TO MAINTAIN ALT; WE TOLD ATC WE HAD TO DSND; AND WE WERE CLRED TO FL350. AN EMER WAS DECLARED; AND THE APU WAS STARTED AND PUT ON THE BUS. THE ENG FAILURE/SHUTDOWN CHKLIST FOLLOWED BY THE INFLT ENG START CHKLIST (STARTED ASSIST) WAS ACCOMPLISHED. FURTHER DSCNT WAS MADE TO FL290 WITH A RELIGHT ON THE L ENG BY APPROX FL310. IMMEDIATE CLRNC WAS GIVEN TO XXX. WE LANDED UNEVENTFULLY. NO ASSISTANCE WAS NEEDED FROM THE CFR CREWS. OUR DECISION TO LAND AT XXX WAS BASED ON SEVERAL FACTORS. WX WAS OUR MAJOR CONCERN. WE HAD JUST COME OUT OF A MODERATE CHOP CONDITION; AND IT TOOK A SIGNIFICANT TIME AND DISTANCE TO GET OUT OF ENOUGH RADAR RETURNS TO EVEN CONSIDER A SAFE LET-DOWN TO POSSIBLY YYY OR XXX WHICH ULTIMATELY LOOKED EQUIDISTANT FROM OUR COURSE OF FLT. SIGNIFICANT WX WAS BELOW US AS WE PASSED OVER OTHER ARPTS HAD A VERY LARGE RED CELL OVER IT. TURNING BACK SBOUND SEEMED OUT OF THE QUESTION AS OUR DISPATCH WX PACKAGE SHOWED WX THERE. THE ROUGH RIDE LESSENED IN THE DSCNT; BUT WE DID NOT GET TO A CONDITION THAT WE EVEN THOUGHT WAS SAFE FOR THE FLT ATTENDANTS TO GET UP AGAIN. TURB WAS OUR SECOND MAJOR FACTOR; ESPECIALLY IN THIS CASE OF LOSING AN ENG DURING A TURBULENT SITUATION. THEREFORE; WE CHOSE TO ATTEMPT TO KEEP 2 GOOD ENGS FUNCTIONING BY AVOIDING ANY CHOPPY CONDITIONS THAT WE THOUGHT MIGHT PRESENT A PROB AGAIN. A SAFE LNDG IN NON-TURBULENT AND NON-RAINY CONDITIONS WAS OUR ULTIMATE GOAL IN SELECTING XXX AS OUR LNDG ARPT. WX AVOIDANCE IS A NEVER-ENDING PROCESS. SUSPECT RETURNS AT HIGH ALTS SHOULD ALWAYS BE QUESTIONED (AS WE DID). HOWEVER; CONTINUOUS TILT CHANGE OF RADAR ELEVATION AND FURTHER QUESTIONING OF WHAT ATC SEES IS OF BENEFIT. ASK ATC ABOUT THE TOPS AND THE LEVEL OF INTENSITY. ALSO; SELECT ENG ANTI-ICE IF SUSPECT WX IS A POSSIBLE PROB.