BE76 INSTRUCTOR PLT AND STUDENT WENT OFF THE SIDE OF THE TKOF RWY; AFTER THE #1 ENG WOULD NOT PRODUCE TKOF THRUST. THE ENG WAS PREVIOUSLY SQUAWKED FOR THE SAME PROB.

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I WAS TO CONDUCT A MULTI-ENG TRAINING FLT WITH A STUDENT. MY STUDENT HAD APPROX 800 HRS OF SINGLE-ENG TIME AND APPROX 4 HRS OF MULTI-ENG TIME. THE PREVIOUS DAY; THE BEECHCRAFT DUCHESS HAD BEEN WRITTEN UP (SQUAWKED) FOR A PROB WITH THE L ENG AS IT WOULD ONLY PRODUCE 2200 OF 2700 RPM. THE ACFT WAS THEN SIGNED OFF AS FIXED BY A QUALIFIED MECH. WE PROCEEDED TO COMPLETE THE START; TAXI; AND RUN-UP CHKLISTS; AND ALL INDICATIONS WERE NORMAL. WE THEN PROCEEDED TO TAXI TO THE ACTIVE RWY AND WERE CLRED FOR TKOF. THE STUDENT WAS PERFORMING THE TKOF; AND AS HE PUSHED THE THROTTLES UP TO 2000 RPM; I CHKED ALL THE GAUGES; HE RELEASED THE BRAKES AND WE PROCEEDED DOWN THE RWY. VERY SHORTLY THEREAFTER I NOTICED THAT WE BEGAN TO DRIFT TO THE L. MY STUDENT QUICKLY PANICKED AND FROZE ON THE THROTTLES. AFTER TELLING HIM NUMEROUS TIMES TO ABORT WITH NO ACTION ON HIS PART; I QUICKLY PULLED BOTH MIXTURES TO THE IDLE CUTOFF POS. UNFORTUNATELY; BY THIS TIME WE WERE QUICKLY APCHING THE EDGE OF THE RWY. DESPITE HVY BRAKING ATTEMPTS BY BOTH STUDENT AND INSTRUCTOR; THE ACFT PROCEEDED TO ROLL APPROX 50-100 FT ON THE GRASS. NO DAMAGE WAS DONE TO THE AIRFIELD; AND WE RESTARTED THE ENGS AND TAXIED BACK TO THE RAMP. THIS WAS A NEEDLESS ACCIDENT. THE ENG APPARENTLY WAS NOT FIXED; AND SHOULD NEVER HAD BEEN PUT BACK IN SVC. OF NOTE; THE ACFT WAS WRITTEN UP THE FIRST TIME BY THE DIRECTOR OF TRAINING AT OUR FLYING SVC; AND HE HIMSELF STATED THAT 'THE MECH PROBABLY JUST RAN IT UP ON THE GND AND SIGNED IT OFF.' THE STUDENT WAS NOT TO BLAME; FOR HE SHOULD NOT BE EXPECTED TO PERFORM CORRECTLY IN THIS SIT. HAD THE INSTRUCTOR NOT BEEN PROPERLY TRAINED AND REACTED QUICKLY; THIS COULD HAVE BEEN A DISASTER! ACFT WRITE-UPS MUST BE TAKEN SERIOUSLY; AND MAINT FLTS MUST BE PERFORMED IF NECESSARY. BECAUSE WE DO OUR RUN-UP AT 2200 RPM; IF THE ACFT IS ONLY RUN-UP ON THE GND; IT WOULD HAVE BEEN IMPOSSIBLE TO HAVE KNOWN IF THE PROB WAS RESOLVED.