ACR MLG CHANGED HEADING WITHOUT PERMISSION OR TELLING CENTER AND NEARLY CONFLICTED WITH ANOTHER ACFT.

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CLIMBING ON AN APPROX HEADING OF 200 DEG BNA DEP ASSIGNED US A 270 DEG HEADING. WE MADE THE TURN. UPON ROLLING OUT BNA GAVE US A FREQ CHANGE TO MEM CENTER; WHICH WE ACKNOWLEDGED. I DID NOT LIKE WHAT I SAW ON THE RADAR AND I BROUGHT IT TO THE CAPT'S ATTN. WE BOTH AGREED A 30 DEG TURN TO THE LEFT WAS THE BEST COURSE OF ACTION. AFTER COMPLETING THE TURN TO 240 DEG THE CAPT CHECKED IN WITH CENTER. BEFORE WE COULD TELL CENTER WE WERE DEVIATING FOR WEATHER; CENTER WAS CHEWING US OUT FOR MAKING THE TURN AND NOT LETTING ANYONE KNOW. HE THEN ASSIGNED US A 290 DEG HEADING AND SAID WE ALMOST FLEW INTO AN LTT. WE MADE THE TURN TO 290 DEG; AS WE HAD NOW PASSED THE CELL. FROM THE TIME WE INITIALLY ROLLED OUT ON THE 270 DEG HEADING TO THE TIME WE WERE GIVEN THE 290 DEG HEADING BY CENTER LESS THAN TWO MINUTES HAD ELAPSED. WE COULD HAVE GIVEN CENTER ABOUT 30 SEC MORE NOTICE OF OUR INTENTIONS BUT WE DID NOT FEEL THAT THE DEVIATION WAS LARGE ENOUGH OR LONG ENOUGH TO BE NOTICED. WITH CELLS IN THE AREA ATC SHOULD ANTICIPATE DEVIATIONS WOULD BE NECESSARY.