A little research shows that it's not all that rare.
First, though, we must remember that the Cy Young Award wasn't instituted until 1956 (Don Newcombe) and didn't become a two-league award until 1967. And the Manager of the Year Award wasn't instituted until 1983. Thus, we can eliminate, for instance, 1968, when Bob Gibson and Denny McLain won both the MVP and the Cy Young Award in their respective leagues; since no Manager of the Year award was issued (one can surmise that Red Schoendienst and Mayo Smith might well have taken the prizes, had they existed).
But, looking at the seasons from 1983 on, we find a number of occasions – five, to be exact - where one team swept these three particular awards:
In 1986, John McNamara won MOY (the vote was taken before Mac left Bill Buckner in the game, of course) and Roger Clemens won MVP and CYA.
The 1984 Cubs did the same thing when Jim Frey won MOY, Ryne Sandberg MVP, and Rick Sutcliffe CYA; so did the 1990 Pirates with Jim Leyland as MOY, Barry Bonds as MVP, and Doug Drabek as CYA. So did the 1991 Braves with Bobby Cox as MOY, Terry Pendleton as MVP, and Tom Glavine as CYA, and the 1992 A's with Tony La Russa as MOY and Dennis Eckersley as both MVP and CYA.
Another sort of trifecta has occurred three times, where a team has won the MOY, the MVP, and the Rookie of the Year - in 1988, the Oakland A's got three – La Russa as MOY, Jose Canseco as MVP and Walt Weiss as ROY.
Likewise, in 1985, Whitey Herzog won MOY, Willie McGee MVP, and Vince Coleman ROY.
And in 2001, Lou Piniella won MOY, and Ichiro Suzuki won MVP and ROY.
So the question isn't so much how rare is it for a team to win three major postseason awards – it's been done eight times; five with Manager of the Year, MVP, and Cy Young Award and three with Manager of the Year, MVP, and Rookie of the Year – but whether a team has ever won all four awards.
And the answer is… never.
Looking at the 1992, 1990, 1986 and 1984 trifectas, neither the 1992 A's, the 1990 Pirates, the 1986 Red Sox, or 1984 Cubs had anyone get even one point in the ROY voting, so they're eliminated from consideration.
The 1991 Braves did get Brian Hunter on the ROY list; he came in fourth to Bagwell, Merced, and Lankford. Strike Five.
In two other cases – the 1988 A's and 1985 Cardinals – the team had a pitcher finish second in the Cy Young Award voting – Dennis Eckersley finished a distant second to Frank Viola in 1988; John Tudor a not-quite-as-distant second to Dwight Gooden in 1985. Strike Six. Strike Seven.
In a third case, Seattle's Freddy Garcia finished third (behind Mark Mulder!) to Roger Clemens in the Cy Young voting. Ironically, Randy Johnson won the National League Cy Young Award that season, but of course he was an ex-Mariner by that time, pitching for the Arizona Diamondbacks. Since Garcia was part of the trade that sent Johnson out of Seattle, it could be argued that had the Big Unit stayed in Seattle, there would have been your clean sweep – and why not? The Mariners won an AL-record (and tied for the major league record) 116 games that season before failing in the postseason. But… Strike Eight. We're out.
So, no team has ever completed the clean sweep of all four awards (and the 2005 Cardinals won't be on that list, either, since Yadier Molina isn't rookie-eligible and Brad Thompson isn't likely to crack the list of top NL rookies).
But, if it's any consolation, I would say the 1985 Cardinals may have been the team that came the closest to winning all four awards in the same year (as Tudor came closer than Eckersley to winning the Cy Young Award), with the possible exception of the 2001 Mariners, who would have had all four had Randy Johnson still been with them.