CAPT OF ACR LGT ACFT DEV FROM THE TRACK OF A SID DUE TO LOOKING FOR TFC WARNINGS OF ACFT TCASII.

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OUR ORIGINAL ATC CLRNC WAS VIA THE MOUNTAIN 2 DEP TO MAINTAIN 9000 FT. ABOUT 4 MI AFTER TKOF FROM RWY 34L; SEATTLE DEP CTL TOLD US TO MAINTAIN 5000 FT DUE TO TFC AT 12 O'CLOCK AT 6000 FT. AT ABOUT THE SAME TIME; THE TCASII CALLED OUT 'TFC; TFC' WHICH WAS FOR ANOTHER TARGET TO THE W OF US. THAT TARGET HAD NO ALT READOUT; THEREFORE WE MADE NO EFFORT TO FIND IT VISUALLY. WE STILL HAD NOT SEEN THE 6000 FT TFC; WHEN THE TCASII SOUNDED 'TFC; TFC' ONCE AGAIN. IN FACT; WITHIN THE NEXT MIN; THERE WERE AT LEAST 4 MORE TCASII PROMPTS FOR OTHER TARGETS. AS IT TURNS OUT; THOSE TARGETS PROBABLY WERE ASSOCIATED WITH BOEING FIELD. WHILE THIS WAS GOING ON; WE DID NOT TURN AT THE 8 DME POINT ON THE DEP; BUT CONTINUED ON AND AT ABOUT THE 13 DME POINT; AS I WAS ABOUT TO START A TURN TO THE E; DEP CTL ASKED IF WE HAD RECEIVED THE MOUNTAIN 2 DEP. WE RESPONDED THAT WE HAD; AND WERE THEN INSTRUCTED TO TURN TO A HDG OF 110 DEGS FOR VECTORS ENRTE. TCASII IS A VERY USEFUL TOOL; AND HAS IN MY EXPERIENCE PLAYED A VERY IMPORTANT ROLE IN FLT SAFETY; HOWEVER THIS TIME IT ALMOST OVERLOADED OUR CREW AND WAS VERY DISTRACTING AT A CRITICAL TIME OF OUR FLT.