AN IFR C303 PLT DESCRIBES HIS DIFFICULTY WHEN HE COULD NOT COMPLY WITH ATC'S CLB REQUESTS BECAUSE OF TSTMS AND TURB AT HIGHER ALTS.

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ON IFR FLT PLAN TO ZZZ VIA ABC. FILED FOR 6000 FT. UPON TKOF DEP CTL CLRED US TO 3000 FT ON A HDG OF 240 DEGS. EVENTUALLY WE WERE CLRED TO 5000 FT BUT WERE UNABLE DUE TO SEVERE TURB ENCOUNTERED AT 3500 FT. WE REQUESTED BACK TO 3000 FT WHICH WAS GRANTED. EVENTUALLY DEP CTL ASKED US TO TURN BACK TO THE E ON A HDG OF 090 DEGS AT 3000 FT. WE EXPLAINED SEVERE TURB ENCOUNTERED AND REQUESTED 240 DEGS OR EVEN FURTHER S AT 3000 FT. DEP CTL ALLOWED US TO HEAD AT 240 DEGS AGAIN AND AT 3000 FT CTL WAS XFERRED TO ZAU WHICH REQUESTED WE CLB TO 6000 FT; WE SAID UNABLE DUE TO TURB ABOVE. ZAU SEEMED UPSET THAT WE WERE ON AN IFR FLT PLAN AND COULD NOT GO ABOVE 3000 FT; WE EXPLAINED THE TURB AND WERE EVENTUALLY XFERRED TO ANOTHER ZAU CTLR WHO WORKED WITH US AROUND THE STORM CELL AND WHEN CLR OF THE STORM ALLOWED US TO GO TO 6000 FT. THE REMAINDER OF THE FLT WAS SMOOTH. CTLRS WERE VERY BUSY ON THIS PARTICULAR DAY. THE RIGHT SOLUTION WAS FOR US TO STAY ON 240 DEG HDG AT 3000 FT UNTIL CLR OF STORM DUE TO SEVERE TURB. MUCH OF THE REMAINDER OF THIS FLT WAS SPENT IN SMOOTH AIR IN IMC AT 6000 FT. CORRECTIVE ACTION WOULD HAVE BEEN FOR THE FIRST ZAU CTLR TO ACCEPT OUR COM OF 'SEVERE TURB' WHICH WAS SO ROUGH OUR HEADS HIT THE CEILING OF THE ACFT AND KNOCKED MY HEADSET OFF. 30 MINS BEFORE THE FLT WE OBTAINED MULTIPLE DUATS BRIEFINGS. GREEN BAY TWR COMMUNICATED SEVERE TURB RPTS FROM OTHER ACFT. EXPECTED TURB WAS THOUGHT TO BE NW OF OUR POS.