The '78 Yankees aren't available. I tried to only do one per era and the '77 Yankees are in but have been claimed. The only Yankees team left is the '61 version which I'm going to give you unless I hear otherwise.
If I can't have the 92 Jays, then I'd like the 1995 Atlanta Braves. They impressed me a lot in the 92 World Series, so after the short season, I went to them as my NL team, thanks to TBS broadcasts.
'05 Giants-John McGraw's (and the NL's) first championship
'06 White Sox-The Hitless Wonders who beat the 116 win Cubs in the World Series
'25 Pirates-The Waner Brother's
'35 Tigers-Mickey Cochrane and the boys beating Dizzy Dean and the Gas House Gang
'82 Brewers-The franchise's only league championship to date-The Harvy Wallbangers
'88 Dodgers-Kirk Gibson's home run against the Eck
'98 Padres-Tony Gwynn's one trip to the World Series
And the round 1 games are out. I screwed up the game numbers but everyone should have the right ones (they just won't correspond to the ones on the bracket page). Any issues, let me know.