CTLR GAVE CLRNC THAT TOOK ACR MLG INTO HOT RESTRICTED AREA AT AN UNUSUALLY LOW ALT.

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TURBULENCE WAS ENCOUNTERED AT CRS FL310. IN SEARCH OF SMOOTHER FLT CONDITIONS THE FLT EVENTUALLY ENDED AT 17000 FT; OTHER AIR CARRIER FLTS AHEAD OF OURS HAD DSNDED TO 15000 FT AND 12000 FT; AND WERE STILL RPTING MODERATE TURBULENCE. MLG X WAS CLRD DIR TO TAYLOR VOR PRIOR TO REACHING 17000 FT. WE WERE OPERATING ON THAT CLRNC WHEN WE DSNDED TO 17000 FT. PRIOR TO OVERFLYING THE TAYLOR VOR; ATC (JACKSONVILLE CTR) ISSUED A TURN TO A 150 DEG HDG FOR TFC. WE WERE THEN INSTRUCTED TO TURN TO A 170 DEG HDG AND INTERCEPT THE GATER 5 ARR INTO FLL. A SECOND TURN OFF COURSE FOR TFC WAS ISSUED AND COMPLIED WITH; THEN WE WERE ADVISED TO REINTERCEPT THE ARR. THE FLT THEN INTERCEPTED AND WAS FLYING ON THE 153 DEG RADIAL OF GAINESVILLE VOR; ENRTE TO GATER INTXN WHEN JACKSONVILLE CTR SAID; 'MLG X; TURN R TO 240 DEGS IMMEDIATELY.' I IMMEDIATELY COMPLIED WITH CTR'S COMMAND. APPROX 30 SECS LATER CTR ISSUED A TURN CLRNC BACK TO THE L TO 155 DEGS TO INTERCEPT THE BISCAYNE BAY VOR 335 DEG RADIAL; AND FLY IT INBND. THEY STATED THAT WE HAD FLOWN INTO AN ACTIVE RESTR AREA THAT WAS IN USE BY THE U.S. ARMY FIRING LIVE ORDINANCE ABOVE 12000 FT FROM THE SURFACE. I QUESTIONED THEM BECAUSE THE 2 90 DEG TURNS IN RAPID SUCCESSION DIDN'T SEEM TO ME TO DISPLACE THE ACFT FAR ENOUGH OVER THE GND TO INDICATE THAT WE WERE IN FACT IN A RESTR AREA. THE RESPONSE WAS THAT RESTR AREA 2901 WAS HOT AND THAT THEY WERE UNAWARE OF IT; BUT THAT THEY HAD GOTTEN US OUT. I MAINTAIN THAT THE FLT WAS BEING CONDUCTED IN ACCORDANCE WITH AN ATC CLRNC; FLT CONDITIONS WERE IN WX; MODERATE TURBULENCE; LIGHT ICING; AND NO GND CONTACT. THE FLT WAS VECTORED BY JACKSONVILLE CTR INTO AN ACTIVE RESTR AREA.