CL65 CREW WAS CLRED TO TAXI ON A CLOSED RWY 31.

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UPON LNDG RWY 4R AT JFK; WE EXITED AT TXWY FB HIGH SPD. THE CAPT TAXIED FORWARD TO CLR US OFF THE ACTIVE; PASSING TXWY FA. GND CTL CLRED US TO TAXI VIA TXWYS FA; Y; J TO THE RAMP. WE WERE JUST FIGURING OUT THAT WE HAD ALREADY PASSED THE FA INTXN WHEN GND INFORMED US THAT WE HAD PASSED IT; AND REVISED OUR TAXI CLRNC TO PROCEED VIA TXWYS FB; Y; CROSS RWY 31R; TXWY J TO THE RAMP. THE CAPT AND I DISCUSSED THIS TAXI ROUTING; AND THEN I PROCEEDED TO COMPLETE THE AFTER LNDG CHKLIST. THE CAPT THEN USED COM #1 TO MONITOR GND CTL; WHILE I USED COM #2 TO CONTACT OUR OPS TO INFORM THEM OF OUR ARR AND RETRIEVE GATE ASSIGNMENT. UPON APCHING RWY 31R; THE CAPT CONFIRMED ON COM #1 CLRNC TO CROSS THE RWY; AND GND CTLR CONFIRMED THIS. AS I LOOKED UP; I NOTICED 3 CONES SPACED ABOUT 2 FT APART; WITH THE CTR CONE ON THE TAXI LINE. I APPLIED THE BRAKES AND SAID; 'CONES. THERE ARE CONES AHEAD;' OVER THE INTERCOM. THE CAPT REPLIED THAT WE WERE CLRED TO CROSS; AND PROCEEDED AHEAD; THINKING THAT I HAD BRAKED TO PREVENT A RWY INCURSION. UPON REACHING THE NW HOLD SHORT LINES OF THE RWY ON TXWY Y; WE SPOTTED 3 MORE CONES; ALSO SPACED NARROWLY ACROSS THE MIDDLE OF THE TXWY. THE CAPT STOPPED AND CALLED GND CTL TO INFORM THEM WE HAD CONES BLOCKING THE WAY. AFTER SEVERAL SECONDS; GND RADIOED BACK STATING THAT HE HAD NOT BEEN INFORMED OF THE CLOSURE OF RWY 31R FOR TAXI BY MAINT; AND WOULD SEND A TRUCK OUT TO CLR THE CONES. MAINT ARRIVED SHORTLY AFTER AND REMOVED THE CONES. WE TAXIED TO THE RAMP VIA TXWYS J; A; LA WITHOUT FURTHER INCIDENT. GND CTL SHOULD HAVE BEEN INFORMED OF THIS MAINT; SO THAT THE TAXI CLRNC WOULD RTE US AROUND THE CLOSED AREA. ALSO; THE CONES USED WERE NOT VERY VISIBLE AND WERE NARROWLY SPACED SUCH THAT UNLESS YOU HAPPENED TO BE LOOKING RIGHT DOWN THE CTRLINE; YOU WOULDN'T SEE THEM. MORE VISIBLE DEVICES USED FOR TEMPORARY CLOSURES WOULD HAVE HELPED AS WELL.