A DC10 CREW; AFTER BEING PUSHED BACK AT ORD; STARTED TAXIING BEFORE THE GND CREW WAS CLR OF ACFT.

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PUSHBACK/AFTER START/GND CREW CLRED OFF TO FO SIDE. WE APPEARED TO BE BLOCKING TXWY FOR OTHER COMPANY ACFT THAT HAD PUSHED BACK BEFORE US. FOCUSED MY ATTENTION ON HIS APPARENT HOLDING ON RAMP FOR US. AFTER NORMAL PERIOD OF TIME FOR GND CREW TO CLR; STATED WE HAVE TO GET OUT AHEAD OF COMPANY. I ASKED FO IF WE WERE CLR R - HE CONFIRMED CLR R BEGAN TO MOVE A/C. AFTER 1-2 INCH OF ROLL FELT/HEARD CLUNK FROM NOSE WHEEL STRUT HITTING TOW BAR. GND CREW CAME BACK ON SAID; 'HOLD BRAKES; HAVING TROUBLE WITH TOW BAR.' NO DAMAGE TO A/C. BROKE PIN ON TOW BAR. I ASSUMED FO UNDERSTOOD GND CREW WAS CLRING TO THE R AND WHEN I ASKED IF CLR IT WAS IN FACT; CLR. HE THE FO HAD MISSED GND CREW; TELLING US 'CLR R' ME ACKNOWLEDGING CLR R AND SAYING; 'OFF ON YOUR SIDE.' I RUSHED HIM WITH STATEMENT ABOUT NEEDING TO CLR RWY FOR COMPANY ACFT. I USUALLY CONFIRM WITH FO THAT CREW IS CLRING R SIDE IF THAT IS THE CASE. I WAS PREOCCUPIED WITH COMPANY ACFT AND DID NOT GET CONCURRENCE GND CREW CLR - JUST ASSUMED HE WOULD SPEAK UP IF NOT CLR.