B757 CREW FLEW A DIFFERENT RTE THAN THE ZME CTLR EXPECTED.

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FLT SEGMENT LIT-SDF. WE CONTACTED CLRNC DELIVERY (TWR) ON THE GND IN LITTLE ROCK. CTLR GAVE US THE CLRNC 'AS FILED; CLB MAINTAIN 4000 FT EXPECT FL330. 10 MINS LATER; 'DEP FREQ 118.7; SQUAWK ABCD.' OUR FILED CLRNC WAS LIT J131 PXV CHERI2 SDF. AS PART OF TKOF CLRNC THE SAME CTLR (ALSO TWR OPERATOR) CLRED US FOR A 'L TURN ON COURSE.' AFTER TKOF; WE MADE A L TURN (RWY 22R) AND JOINED J131 TO PXV AS FILED. THE TWR CTLR WAS ALSO THE DEP CTLR AND HE PASSED US OVER TO ZME; AS WE WERE CLBING THROUGH APPROX 11000 FT; ON OUR WAY TO 15000 FT. WE CHKED IN WITH ZME AND RECEIVED CLRNC TO FL230. SEVERAL MINS LATER WE RECEIVED CLRNC TO FL330 AS FILED. PASSING FL270 MEMPHIS ASKED US 'WHERE ARE YOU GOING?' WE RESPONDED WITH OUR FILED CLRNC; LIT J131 PXV CHERI2 SDF. MEMPHIS REPLIED 'THAT IS NOT YOUR CLRNC. YOUR CLRNC IS BNA EWO SDF.' WE EXPLAINED THAT THE CLRNC DELIVERY/TWR OPERATOR/DEP CTLR CLRED US 'AS FILED.' MEMPHIS SAID 'THAT CLRNC IS INCORRECT' AND THAT HE WAS GOING TO RPT IT. MEMPHIS ALLOWED US TO CONTINUE COURSE TO PXV THEN THE CHERI2 TO SDF. ON THE GND IN SDF; WE LEARNED THAT ZID CLOSED THE PXV ARR GATE TO SDF SEVERAL HRS PRIOR TO OUR DEP FROM LIT BECAUSE OF A RWY CLOSURE AT SDF. MEMPHIS KNEW OF THE ARR GATE CLOSURE AND OUR SUBSEQUENT REROUTE OVER BNA (NASHVILLE). NORMALLY; LIT SHOULD HAVE REQUIRED A COMPLETE CLRNC READBACK BECAUSE OF THE REROUTE. CLRNC DELIVERY AT LIT NEVER INFORMED US OF THE REROUTE; AND THEY FAILED TO READ OUR CLRNC COMPLETELY.