Is there any modern music for the harpsichord that doesn’t use or imitate old music forms? Especially any modern-sounding harpsichord concertos?
The best-known one may be Poulenc’s Concert Champetre.
There’s a great, Stravinskyish concerto for harpsichord and flute, oboe, clarinet, violin, and cello by Manuel de Falla. The version I have, with my one-time music professor Anthony Newman on harpsichord, appears to be out of print (and is on a CD with otherwise non-harpsichord works), but is available used on Amazon. There’s another recording out there paired with harpsichord music by Martinu, which might also be up your alley, though I’m not familiar with it myself.
Before he passed away in the mid-80’s my composition instructor was doing A=420 pieces in modern styles for a Baroque chamber group. I’m sorry I can’t remember much more than that. The composer’s name is Eric Heckard – I don’t remember the ensemble.
There are several pieces by György Ligeti that fit the bill: Continuum, Passacaglia Ungharese, and Hungarian Rock come to mind. Also, Elliott Carter has a Double Concerto for Piano and Harpsichord, and a Sonata for Flute, Oboe, Cello, and Harpsichord. Also I think Bohuslav Martinů wrote a concerto for haprsichord.
Here’s a list that includes just well-known composers. In actuality, there are hundreds, if not thousands, of modern and contemporary harpsichord works. There are a few concertos in here.
Samuel Adler:
Sonata for harpsichord
Sonata for violin & harpsichord No. 2
Albright, William
4 Fancies for harpsichord
Andriessen, Louis
Overture to Orpheus
Berio, Luciano
Rounds
Berkeley, Sir Lennox
Suite for harpsichord
Cowell, Henry
Set of Four
Distler, Hugo
Harpsichord Concerto, Op. 14
Glass, Philip
Harpsichord Concerto
Goeyvaerts, Karel
Litanie 5
Gorecki, Henryk
Concerto for harpsichord
Gubaydulina, Sofiya
Ritorno perpetuo
Harrison, Lou
Gothic Piece
Harpsichord Sonata
6 Sonatas for Cembalo
Hovhaness, Alan
Sonata 1 for harpsichord, Op. 306
Sonata 2, for harpsichord, Op. 318
Sonata 3 for harpsichord, Op. 336
Sonata 4 (“Daddy-long-legs”), for harpsichord, Op. 357
Sonata 5 for harpsichord, Op. 361
Ligeti, Gyorgy
Continuum, for harpsichord
Hungarian Rock (Chaconne), for harpsichord
Passacaglia Ungherese, for harpsichord
Martin, Frank
Harpsichord Concerto
Martinu, Bohuslav
Harpsichord Concerto
Harpsichord Sonata
2 Impromptus for harpsichord
2 Pieces for harpsichord
Rodrigo, Joaquin
Preludio y ritornello
Rorem, Ned
Spiders
Schnittke, Alfred
3 Fragments
Shchedrin, Rodion Konstantinovich
Stalin Cocktail
Takemitsu, Toru
Rain Dreaming
Xenakis, Iannis
Khoaï
Komboï
Naama
Young, La Monte
Fugue for organ or harpsichord in C minor
Zwilich, Ellen Taaffe
Fantasy for Harpsichord
There’s lots more where that came from 
The harpsichord enjoyed a very brief favor around 1940 in the Alec Wilder Octet (where Walter Gross, a studio musician played it), which played Wilder’s impressionistic, arranged compositions with a strong pop-jazz flavor. It was also part of the Artie Shaw Gramercy Five (where it was played by noted jazz pianist Johnny Guarnieri), which was more of a straight-ahead swing group.
Thoroughly recommended 
Also, in addition to the Schnittke mentioned, lots of his other pieces use the instrument. Although it’s often referential or in parody of older idioms, it’s certainly not anything other than 20th-century music.
The thread title made me think of George Gruntz and an album of his (Jazz Goes Baroque) that I bought back in the 60’s. It’s a jazz thing and until Beware of Doug posted about jazz, I thought it best to keep quiet. There’s also an early Henry Mancini album (Combo!) featuring harpsichord played by John (Johnny) Williams (yes, that John Wiliams), which is still sort of jazzy.
That’s a lot more than I thought. Thank you, everyone, especially Figaro.
Frank Martin’s Petite Symphonie Concertante uses harpsichord, although other instruments are featured too.
On the rock/pop side of things, several of The Kinks’ songs* from the mid-to-late 60’s feature harpsichord.
*titles provided upon request