ACR X TCASII WITH 3 ACFT HAD LTSS FROM ACR Y. EVASIVE ACTION TAKEN.

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ACR X ON DSCNT INTO TYS WAS DSNDING TO 4000 FT. LEAVING APPROX 5800 FT; CTLR GAVE US A 150 DEG HDG FOR FINAL. WE NOTICED A TCASII TA; OPPOSITE DIRECTION; CLBING THROUGH OUR ALT APPROX 1000 FT BELOW. AT THE TIME WE NOTICED HIM; I PICKED HIM UP VISUALLY AND COULD SEE IT WAS GOING TO BE CLOSE. I INCREASED MY RATE OF TURN TO AVOID ONCOMING TFC; AN MLG. WE PASSED WITHIN ABOUT 1 1/2 - 2 MI OF EACH OTHER HORIZLY AND 300-400 FT VERTICALLY. WE THEN WERE ON A 150 DEG HDG AND WERE GIVEN WHAT I THINK WAS A FURTHER DSCNT. AGAIN; TFC SHOWED UP OPPOSITE DIRECTION AND CLBING. OTHER TFC WAS AN ACR Y WHO WAS QUESTIONING THE CTLR ABOUT HIS RA. AT THIS TIME WE RECEIVED AN RA TO CLB AT 1500 FPM TO AVOID TFC. WE PASSED WITHIN 200 FT VERT; UNKNOWN HORIZ. WE NEVER SAW THIS AIRPLANE. WE WERE IN THE TOPS OF THE CLOUDS. HE WAS IN THE CLOUDS; I ASSUME. HE STATED TO ATC THAT HE DID NOT SEE US. I BELIEVE WITHOUT TCASII WE WOULD HAVE COME EXTREMELY CLOSE IF NOT HIT. WE WERE NOW CLR OF TFC AND AT 6000 FT DSNDING ONCE AGAIN. WHEN WE BROKE OUT OF THE CLOUDS WE WERE SE OF FINAL HDG SE. CTLR GAVE US A HDG FOR FINAL AND CLRNC TO 2500 FT. WE COULD SEE THE ARPT WHICH WE ADVISED ATC. CTLR SAID TO FLY DIRECT TO ARPT. WE THEN ASKED IF THAT MEANT WE WERE CLRED FOR A VISUAL. CTLR SAID NO BECAUSE OF TFC; AN ACFT Z. THE TFC WAS NOW AT OUR 12 O'CLOCK; 600 FT BELOW. WE WERE AGAIN GIVEN A HDG THAT HAD TAKEN US TOWARD ANOTHER ACFT. WE WERE DSNDING INTO THIS VFR ACFT Z AND RECEIVED ANOTHER RA TO CLB. A NEW CTLR CAME ON AND TOOK OVER TO VECTOR US AWAY FROM ARPT TO ALLOW US TO DSND AND GET BACK ON A NORMAL APCH. AFTER HAVING 3 RA'S IN LESS THAN 5 MINS; I WAS A LITTLE SHAKEN.