BE76 INSTRUCTOR PLT AND STUDENT PLT HAD A RWY INCURSION OF RWY 36R AT DFW.

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THE STUDENT AND I HAD ADVISED GND CTLRS AT DFW SEVERAL TIMES THAT WE WERE UNFAMILIAR WITH THE ARPT. AFTER LNDG ON RWY 31R; WE WERE INSTRUCTED TO TAXI VIA TXWY R AND THEN L ON TXWY Z; THEN OVER THE TXWY Z BRIDGE. WE CHANGED GND CTL FREQS ATOP THE BRIDGE; AS WE WERE TOLD. THE NEXT CTLR ASKED US TO TURN L ON G11 TO GO TO GA AREA FOR PARKING. AS WE ROLLED OFF THE BRIDGE; I SAW TXWY F; BUT NOT TXWY G OR G11. I ASSUMED G11 WAS STILL TO COME; AND WE APCHED RWY 36R. THINKING G11 WAS STILL COMING AND THAT WE WERE CLRED ACROSS ALL RWYS TO GET TO G11; I TOLD THE STUDENT TO TAXI ACROSS THE RWY. AS WE FINISHED XING THE RWY; THE GND CTLR ASKED OUR POS TO THE RWY. I RESPONDED THAT WE HAD CROSSED IT. HE THEN GAVE ME A PHONE NUMBER TO CALL; AND I CALLED IT UPON PARKING AT GA. I FEEL THAT THIS INCIDENT COULD HAVE BEEN PREVENTED HAD THE STUDENT AND I KNOWN WE WERE TO TURN L ON TXWY G AND THEN TAKE THAT TO G11. THE CTLR WAS NOT SPECIFIC ABOUT THE 'MIDDLE STEP' OF L ON TXWY G TO GET TO G11. MY STUDENT LATER SAID HE DID SEE TXWY G; BUT WAS UNAWARE WE WERE SUPPOSED TO TURN L THERE. WE ALSO SHOULD HAVE SLOWED OUR TAXI AND SHOULD HAVE BEEN FOLLOWING THE TAXI DIAGRAM MORE CLOSELY. WE ALSO SHOULD HAVE ASKED THE CTLR IF WE WERE CLRED ACROSS RWY 36R INSTEAD OF ASSUMING THAT WE WERE. ALL IN ALL; BETTER COM BTWN US AND GND CTL WOULD HAVE HELPED THE SIT; ESPECIALLY AS IT WAS MY FIRST TIME AT DFW AS PIC. SUPPLEMENTAL INFO FROM ACN 632255: THERE WAS CONFUSION OVER THE INSTRUCTION; AND WE RECEIVED IT VERY CLOSE TO THE INTENDED TURN WHICH WAS ON TXWY G. I SAW THE TXWY G MARKINGS; BUT MY INSTRUCTOR DID NOT. BELIEVING THE TURN WAS YET TO COME; WE HAD CROSSED THE HOLD SHORT LINE FOR RWY 36R BEFORE WE REALIZED IT. INCREASING SPD; WE GOT OFF THE RWY AS QUICKLY AS POSSIBLE; AT WHICH POINT WE REALIZED; AND WERE INFORMED BY GND CTL THAT WE HAD MADE A MISTAKE.