(1) To start, the first series to talk about is actually a 9-novel series, John Galsworthy's The Forsyte Chronicles, which is a trilogy of trilogies, each consisting of 3 main novels and long short-story/novella-like "interludes" in the first 2 trilogies:
(a) The Forsyte Saga:
- i. The Man of Property
- Interlude: "Indian Summer of a Forsyte"
- ii. In Chancery
- Interlude: "Awakening"
- iii. To Let
(b) A Modern Comedy
- iv. The White Monkey
- Interlude: "A Silent Wooing"
- v. The Silver Spoon
- Interlude: "Passers By"
- vi. Swan Song
(c) End of the Chapter
- vii. Maid in Waiting
- viii. Flowering Wilderness
- ix. One More River

Reading all 9 is very much in the Masterpiece Theatre mode of British literature, straightforward linear narratives in the "old tradition".  It's ironic that I only read it because I had come across a copy of The Forsyte Saga in a used bookstore, and I had remembered that there was a classic BBC version from the 1960's, which also aired on PBS, with Eric Porter and Kenneth More among the cast.  I hadn't actually seen it, mind you.

The most famous set is indeed the first, because of the characters of Soames Forsyte, "the man of property" and rather a cold fish, in contrast to his younger brother, Young Jolyon, a more bohemian artsy-type.  Soames marries Irene Heron, but the marriage is a disaster.  Eventually Irene leaves Soames and marries Young Jolyon (his third, as opposed to her second).  They have a daughter, Jon.  Soames rebounds and marries a French woman, Annette Lamotte.  They in turn have a daughter, Fleur.

With time, Jon and Fleur eventually meet, and fall for each other big time.  The fact that they're cousins aside, you can imagine the reaction of their respective parents.  The relationship between Jon and Fleur is worked through over the rest of the first trilogy, and through the second trilogy.

Fleur eventually marries an aristocrat, Michael Mont.  The last trilogy focuses on a cousin of Mont's, Dinny Cherrill, with Fleur much more in a supporting role.  The (melo)drama of the conflicts and tensions present with Soames, Irene, Young Jolyon, Jon and Fleur are pretty much dissipated in the third trilogy.  If you're interested in a time capsule of the first quarter or so of upper-middle class British society, it goes down easy enough.  But there's a reason that dramatizations like the 1967 series and the much later 2002 series with Damian Lewis, Gina McKee, Rupert Graves and Ioan Gruffudd focus on the earlier trilogy.

(2) John Dos Passos U.S.A. Trilogy, which consists of:
(a) The 42nd Parallel (1930)
(b) 1919 (1932)
(c) The Big Money (1936)

I read this in high school in 11th grade for a book report, because we each had to choose a novel on our own and write about it.  I wanted to choose an author that I figured no one else would read, and Dos Passos qualified.  In addition to the fictional characters in the trilogy, Dos Passos interjected his portraits of historical figures and events, in a collage of styles, very much in the spirit of film (with hindsight).  I had no idea of Dos Passos' politics at the time, even if I was vaguely aware that the books leaned left in political sympathies, in its portraits of the struggling working class and those who chase or are part of "the big money".  Just to give an idea, at the end of 1919, Dos Passos lets loose with his portrait of the "unknown soldier" from World War I (excerpt), at the end and after his death:

"....and I dropped the tin hat and the sweaty pack and lay flat with the dogday sun licking my throat and adamsapple and the tight skin over the breastbone.

The shell had his number on it.

The blood ran into the ground.

The service record dropped out of the filing cabinet when the quartermaster sergeant blotto that time they had to pack up and leave the billets in a hurry.
The identification tag was in the bottom of the Marne.

The blood ran into the ground, the brains oozed out of the cracked skull and were licked up by the trenchrats, the belly swelled and raised a generation of bluebottle flies,
and the incorruptible skeleton,
and the scraps of dried viscera and skin bundled in khaki

they took to Chalons-sur-Marne
and laid it out in a neat pine coffin
and took it home to God's country on a battleship
and buried it in a sarcophagus in the Memorial Amphitheater in the Arlington National Cemetery
and draped the Old Glory over it
and the bugler played tabs
and Mr. Harding prayed to God and the diplomats and the generals and the admirals and the brasshats and the politicians and the handsomely dressed ladies out of the society column of the Washington Post stood up solemn
and thought how beautiful sad Old Glory God's Country it was to have the bugler play taps and the three volleys made their ring.

Where his chest ought to have been they pinned
the Congressional Medal, the D.S.C., the Medaille Militaire, the Belgian Croix de Guerre, the Italian gold medal, the Vitutea Militara, sent by Queen Marie of Rumania, the Czechoslovak war cross, the Virtuti Militari of the Poles, a wreath sent by Hamilton Fish, Jr., of New York, and a little wampum presented by a deputation of Arizona redskins in warpaint and feathers.  All the Washingtonians brought flowers.

Woodrow Wilson brought a bouquet of poppies."

Citation: John Dos Passos, 1919, pp. 379-380 (Mariner Books edition).

Dos Passos did write a later trilogy, District of Columbia, which I have not read:
(a) Adventures of a Young Man (1939)
(b) Number One (1943)
(c) The Grand Design (1949)

In addition, later in life, Dos Passos turned way right, from defending Sacco and Vanzetti to being pretty much a McCarthyite.  George Packer wrote in this 2005 New Yorker article:

"....after Spain he began a rightward drift, which by the 1964 election had become so extreme that Edmund Wilson wrote him, 'I feel obliged to tell you that your article about the San Francisco convention sounded like a teenager squealing over the Beatles. What on earth has happened to you? How can you take Goldwater seriously?' (Even during his Goldwater phase, though, Dos Passos never repudiated his belief in the Spanish Republican cause.)"

(3) David Lodge's "academic trilogy", although it's not been compiled as such, to my knowledge:
(a) Changing Places (1975)
(b) Small World (1984)
(c) Nice Work (1986)

In Changing Places, the focus is on two academics, Philip Swallow from the University of Rummidge (UK), and Morris Zapp, of Euphoric State University in Esseph (West Coast, USA; think about the name after you pronounce it - took me a while to get it), who do a faculty exchange at each other's universities, leaving their wives behind for the year.  Just to give you an idea, the novel's title and subtitle are:

Changing Places: A Tale of Two Campuses

I'll let you dissect the several levels of literary punnery there.  Suffice it to say that Swallow and Zapp exchange more than jobs in their respective sojourns overseas.  Also, FWIW, Morris Zapp is reportedly based on Stanley Fish.

In Small World, set about 10 years later, Lodge satirizes the highly inbred world of literary academia, with characters who all know each other, trash each other's work, have affairs with each other, meet each other separated by barely one degree of Kevin Bacon, all with lots of literary references and in-jokes (not all subtle) for literary nerds.  For example, the hero is a young literary scholar named Persse McGarrigle, who is young, naive, and, of course, a virgin.  One other character is a washed-up old literary scholar named Arthur Kingfisher, impotent in every way possible.  Their paths do cross at the end, at a literary conference, where they play out the roles of their namesakes in Arthurian legend.  One other character is an Italian "limousine Marxist" scholar named Fulvia Morgana (as in Morgan Le Fay).  Swallow and Zapp are not the centers of attention as in Changing Places, but are certainly among the principals.

In Nice Work, set several years after Small World, the main characters are Robyn Penrose, a feminist scholar at Rummidge, and Vic Wilcox, a factory manager, who meet up as part of a joint industry-academia project where the two sides nominally are to learn from each other.  Of course, you can guess what happens between the two.  The main literary conceit of the novel, though, is that Lodge transplants the 19th century genre of the British industrial novel (e.g. Dickens' Hard Times, Elizabeth Gaskell's North and South, Disraeli's Sybil, or the Two Nations, with quotes from each to head the chapters).  Swallow is the head of the department, and clearly aging (his hearing loss is a running joke), but is very much a supporting character, while Zapp has a very brief cameo in the last section as someone who potentially offers Robyn Penrose an interview for a post on the Euphoric State faculty.

Other, less obscure literary trilogies beside these that I've read are:
(1) Dante Aligheri's La Divina Commedia
(2) J.R.R. Tolkien's The Lord of the Rings
(3) Douglas Adams' The Hitchhiker's Guide to the Galaxy

On the last, I had the good fortune to hear Douglas Adams live once in a university lecture (unfortunately, I didn't have my copy of the trilogy for him to sign).  The introduction included a joke to the effect that "Douglas Adams has the distinction of being the author of the only 5-volume trilogy in world literature".  FWIW, he actually mentioned his favorite of his books at the time as not The Hitchhiker's Guide, but Last Chance to See, his book about endangered species.  

A few book/literature-themed diaries on DK in the last week include:
(1) SusanG has this review of Matthew Stewart's The Management Myth.
(2) SensibleShoes has this latest Write On! entry, on the theme of suspense.
(3) Vita Brevis mourns the end of the PBS show Reading Rainbow.
(4) Shaviv discourses on Fritz Leiber's The Wanderer.
(5) Sarge in Seattle has this diary on Jill Richardson's book tour of Recipe for America.
(6) Admiral Naismith has this Bookybook posting for the month of August.
(7) plf515's has the latest What Are You Reading? up today.

I should finally note that this diary is actually a re-posting from the time a few Wednesday ago where I had intended to cover for cfk, but I got delayed and she put one up a few minutes before I actually got to post this (which managed to get some nice comments before I deleted it).  So this is the second go-around for this one.  With that, the forum below is yours, about trilogies, other books you've read, all things literary.