B737-300 FLT CREW HAS A HDG DEV ON DEP FROM MDW.

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WHILE TAXIING OUT TO MDW RWY 13C; GND ADVISED US OF A RERTE AND TOLD US TO CONTACT CLRNC DELIVERY. MASSIVE CONFUSION ON GND CTL; TWR; AND CLRNC DELIVERY BECAUSE OF RERTES; STOPS FOR W DEPS; ATC COMPUTER FAILURES; AND SHIFT CHANGEOVER. EVENTUALLY GOT NEW ROUTING FROM CLRNC WITH SOME UNFAMILIAR NAVAIDS. WHILE PLUGGING NEW ROUTING INTO BOX; WE WERE ADVISED NEXT FOR TKOF. WE WERE CLRED ONTO HOLD RWY 13C; THEN GOT THE FOLLOWING CLRNC; 'COMPANY NUMBER; L TURN; L TURN ALL THE WAY AROUND TO NINE ZERO; CLRED FOR TKOF.' AS THE PF; I DIALED 090 DEGS IN HDG WINDOW; ADVANCED THROTTLES FOR TKOF. AS WE STARTED ROLLING; I MENTIONED THAT 090 DEGS WAS ONLY 40 DEGS L. IN MY MIND; I WAS SOMEWHAT CONFUSED BY; 'ALL THE WAY ROUND.' I DID NOT ASK FOR CLARIFICATION AS I SHOULD HAVE. AFTER LIFTOFF; TURNED L TO 090 DEGS. UPON CHANGING TO DEP; THEY ASKED US WHAT OUR INITIAL FIX WAS. CAPT RESPONDED WITH POLO. DEP GAVE US A R TURN TO 270 DEGS. NO CONFLICTS WERE NOTED AND CTLR DID NOT MENTION ANY PROBS. UPON LEVELOFF; WE DISCUSSED IT AND DECIDED THAT TWR MUST HAVE MEANT 'ALL THE WAY AROUND TWO NINE ZERO.' I THINK SEVERAL FACTORS CONTRIBUTED TO THIS. 1) TASK SATURATION -- MYSELF AND CTLRS. BTWN RERTES; COMPUTER ISSUES; FMC LOADING; RADIO SATURATION; MENTALLY I WAS BEHIND. I SHOULD HAVE ASKED ABOUT THE HDG AND ALSO SLOWED THINGS DOWN IF I WASN'T READY. 2) TERMINOLOGY -- IN MY MIND; I'M USED TO HEARING; 'HDG 290 DEGS.' HOWEVER; I DIDN'T HEAR; 'ROUND TO 090 DEGS' AS I'M ALSO USED TO HEARING. ONCE AGAIN; WHEN CONFUSED; I SHOULD HAVE ASKED.