A319 WAS FORCED TO DEVIATE FROM ATC ASSIGNED ROUTING TO AVOID WX AND TURB ENCOUNTER WHILE ON CONGESTED ZHU FREQ.

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WE WERE IN LIGHT TO MODERATE CHOP THROUGHOUT THE CLBOUT PHASE OF FLT ASSOCIATED WITH HIGH WINDS AND MILD CONVECTIVE ACTIVITY. WE DID NOT PAINT ANY RED OR YELLOW CELLS THROUGHOUT THE CLB. WE WERE INITIALLY VECTORED OUT OF IAH AND IN THE LOW 20'S; GIVEN DIRECT TO LOA TO RESUME THE SID. WE ASKED WHEN THE RIDES WOULD IMPROVE AND CTR REPLIED 'MUCH HIGHER.' AFTER BREAKING OUT OF A CLOUD DECK IN THE MID 20'S; WE VISUALLY SAW A RAGGED CLOUD FORMATION AT OUR 12 THROUGH 2 O'CLOCK POS (DARK OUT WITH BRIGHT MOON). AGAIN THERE WAS NOTHING BEYOND GREEN ON THE RADAR; BUT WE WERE IN MODERATE CHOP AND KNEW THE CLOUD FORMATION WAS NOT STABLE. WE ASKED FOR A L TURN TO COMPLETELY AVOID THE CLOUD MASS AND CTR REPLIED 'NO L TURNS. AS FAR AS YOU WANT TO THE R.' AFTER STARTING INITIALLY TO THE R; THE CAPT FELT IT WAS BEST TO STAY STRAIGHT AHEAD ON OUR DIRECT LOA HEADING (WE WOULD HAVE HAD TO DO AT LEAST A 90 DEG TURN R TO AVOID THE WORST PART OF THE CLOUD FORMATION). THE CAPT FLEW THROUGH THE ONLY CLR PART OF THE CLOUD FORMATION JUST TO THE L OFF COURSE. BY MISSING CLOUDS; WE STILL ENCOUNTERED MODERATE TURB FOR APPROX 1 MIN. IN ALL; WE WENT NO FURTHER THAN 1 MI L OFF OUR DIRECT HEADING TO LOA TO AVOID THE CLOUD MASS. DURING THIS TIME OF MODERATE TURB; CTR CALLED US AND WE MISSED THE CALL. AFTER CALLING US AGAIN; WE WERE AGAIN BACK ON A DIRECT LOA HEADING AND RPTED THE TURB INCIDENCE AT THAT TIME TO ATC AND TO DISPATCH. DUE TO THE SHORT TIME OF THE L 'JINK' AND RADIO CONGESTION; WE WERE NOT ABLE TO TELL ATC WE WERE JINKING L. WE DID NOT FEEL A SUSTAINED DEV TO AVOID THE CLOUD MASS OCCURRED AND CTR DID NOT IMPLY THAT EITHER; BUT WE DID MISS A RADIO CALL DURING THE TURB ENCOUNTER AND FELT ATC'S FRUSTRATION IN THEIR EFFORT TO CONTACT US. WE FELT A R TURN INTO THE CLOUD MASS WOULD HAVE RESULTED IN A WORSE TURB SIT. CTR REVEALED THAT HE HAD NO PREVIOUS RPTS OF MODERATE TO SEVERE TURB; SO I DON'T FEEL LIKE HE PUT US IN A BAD SPOT NEGLIGENTLY. THE FLT ATTENDANTS AND PAX WERE DOWN THE ENTIRE TIME OF THE INCIDENT. NO ONE WAS INJURED; BUT WE FOUND OUT LATER THAT A FEW PAX DID GET SICK DUE TO THE RIDE. THE CAPT'S FRUSTRATION WAS THAT WE HAD LITTLE OPTION ONCE RESTR TO R ONLY TURN (INTO THE HEART OF THE CLOUD). THE MODERATE TURB MAY HAVE BEEN UNAVOIDABLE. WE KNEW OF THE TURB POSSIBILITY AND THE MILD CONVECTIVE ACTIVITY ENHANCED THAT CHANCE. THE AIRSPACE WAS VERY BUSY AND THE CAPT FLEW THE AIRPLANE THROUGH A PART OF THE CLOUD THAT LIKELY MINIMIZED THE TURB INCIDENT.