POTENTIAL CONFLICT DEVELOPS AFTER FREQ CHANGE MISSED BY 1 ACR ACFT.

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WHILE AT CRUISE FL310 NEAR ROSEWOOD VOR AND ON A DIRECT RTE TO SLN; ZID ASKED US TO MAKE A 15 DEG HDG CHANGE FOR TFC. AFTER SEVERAL MINS ON THE NEW HDG; ZID ASKED US TO TURN TO 180 DEGS. DURING THE TURN; CTR ASKED US TO EXPEDITE THE TURN AND IN DOING SO; THE ACFT DID BUFFET. THIS WAS FOLLOWED BY A TCASII RA WARNING TO CLB AND WE LEFT FL310 FOR FL317. CTR ASKED IF WE HAD CLBED OUT OF FL310 AND WE RESPONDED THAT WE HAD CLBED AS PER THE TCASII. CTR ASKED US TO RETURN TO FL310 WHICH WE DID AND; SHORTLY THEREAFTER; WE WERE AGAIN CLRED TO SLN AND CHANGED CTR FREQS. THE 3 OF US DISCUSSED THIS SIT AND WE COULD NOT SEE WHERE WE HAD ERRED AT ALL. WRONG! WE HAD BEEN CLRED TO TURN L 15 DEGS; ACKNOWLEDGED THE SAME; BUT THEN WE TURNED 15 DEGS R AND NO ONE CAUGHT THE ERROR. EXPLANATION: IN DISCUSSION LATER; ALL 3 OF US THOUGHT IT WAS A R TURN BECAUSE WE HAD JUST COMPLETED A 15 DEG R TURN IN ORDER TO CLB FROM FL280 TO FL310. THERE WERE 2 ACFT ON OUR L AT FL310. THE CONFLICTING TFC WAS AT OUR 1 O'CLOCK POS; CLBING TO OUR ALT; ON ALMOST A 90 DEG INTERCEPT. WE HAD NOT BEEN ADVISED OF THIS ACFT AND DID NOT KNOW HE WAS THERE UNTIL THE TCASII WARNING. WHY WAS THIS ACFT NOT ON OUR FREQ EARLIER AND ALL 3 OF US (CTR AND 2 ACFT) NOT ON THE SAME COM LOOP? PROB: ZID WAS VERY BUSY; MAYBE CONGESTED AT THIS TIME OF THE DAY; AND AFTER HEARING A PORTION OF THE CTR TAPE; I KNOW THAT SOME OF THE CTR'S XMISSIONS WERE NOT REACHING OUR ACFT OR SOME OTHER ACFT. ZID DID NOT DISCUSS THIS INCIDENT WITH US AT ALL ON THE RADIO. WE DID HEAR CTR TALKING TO ANOTHER ACFT AND ASKING HIM WHY HE DID NOT RESPOND TO HDG CHANGES AND THE PLT SAID HE HAD ANSWERED TWICE! RECOMMENDATION: I THINK WE HAVE ALMOST REACHED A POINT OF SATURATION ON THE AIRWAYS DURING PEAK HRS. THE CTLRS DO A GREAT JOB BUT ARE WE ASKING TOO MUCH FROM THEM AT CERTAIN TIMES? LET'S SPEND SOME MONEY AND UPDATE THE SYS AS NEEDED IN THE NAME OF SAFETY. THE NEW JARGON ON THE RADIO MUST BE STOPPED -- 'YOU'RE STEPPED ON;' 'YOU'RE WALKED ON' OR 'BLOCKED; BLOCKED.' I HAVE BEEN FLYING 35 YRS AND I THINK IT IS TIME WE TAKE A STEP BACK AND TAKE A HARD LOOK AT WHERE WE ARE HEADED IN THIS INDUSTRY. SUPPLEMENTAL INFO FROM ACN 274195: IN CRUISE FLT AT FL310 ALL JUST SO. OF ROD A TCASII CONTACT CAME UP ON OUR DISPLAYS AT OUR 11 O'CLOCK POS MOVING QUICKLY FROM TA TO RA AS WE LOOKED OUT AND SPOTTED A DC10 COMING RIGHT AT US; CO-ALT. THE TCASII COMMANDED A DSCNT. I DISCONNECTED THE AUTOPLT WHILE THE FO ATTEMPTED TO GET THE CTLR'S ATTN. IT TOOK SEVERAL ATTEMPTS BEFORE SHE RESPONDED; AND WHEN SHE DID; IT WAS WITH A FREQ CHANGE. BEFORE SWITCHING I EXPRESSED OUR FRUSTRATION AND ASKED WHY SHE DIDN'T SEE THIS CONFLICT DEVELOPING ON HER RADAR. SHE REPLIED THAT WE WERE OFF HER SCOPE AND IN THE NEXT AREA. I REPLIED THAT PERHAPS A MORE TIMELY FREQ CHANGE WOULD HAVE HELPED. WE HAD DSNDED TO FL300 IN RESPONSE TO THE RA. WHEN CLR OF THE CONFLICT WE CLBED BACK TO FL310. ON THE NEXT FREQ (ZID) WE WERE GIVEN A DSCNT TO FL290 AND A TURN TO AVOID ANOTHER CONFLICT. THIS CTLR WAS VERY BUSY AND HAD NO TIME TO DISCUSS THE NEAR MISS EVENT. UPON ARR AT BNA; I CALLED ZID AND WAS TOLD THAT THEY TRIED CALLING US TO WARN US OF THE CONFLICT BUT SINCE WE HADN'T BEEN SWITCHED TO THEIR FREQ WE DID NOT HEAR THEIR CALLS. A NEAR MISS RPT WAS FILED VIA ATC REPRESENTATIVE. SUPPLEMENTAL INFO FROM ACN 273790: AN RA OCCURRED SHORTLY AFTERWARD COMMANDING A DSCNT WITH WHICH WE COMPLIED; DSNDING TO APPROX FL300 BEFORE RETURNING TO CRUISE ALT. THE OTHER ACFT APPEARED TO BE IN A L TURN AND PASSED ABOUT 300-500 FT ABOVE AND ABOUT 3-5 MI IN FRONT OF OUR ACFT. APPARENTLY ZID HAD ATTEMPTED TO CONTACT US SEVERAL TIMES PRIOR TO ZOB'S INSTRUCTION TO CHANGE FREQS.