DDG Accepted

The only serious way to challenge a gambit is to accept it. After 1.e4 e6 2.d4 d5 3.c4, brave fighters take the offered pawn without hesitation: 3...dxe4!. Naturally, 3...dxc4 would transpose to the Queen's Gambit Accepted; all other moves form the theory of the DDG Declined.
W

4.Nc3

This is natural and by far the most played move. It may look like White is intending 5.Nxe4, but that is hardly ever the case.

4.Be3 (Alapin-Diemer-Duhm)

The Alapin-Diemer Gambit (ADG) games where the good old c4 is played later transpose to this variation. Especially the knight on d2 makes this variation unique; other lines are dealt with the 4.Nc3 Nf6 5.Be3 variation. I am a bit reluctant to the Nd2 lines, but DDGers cannot be purists: the whole DDG is just a bunch of transpositions. The following games and analysis are mainly from Tim Sawyer's Alapin French - Tactics for White.

1.e4 e6 2.d4 d5 3.Be3 dxe4 4.f3 exf3 5.Nxf3 Nf6 6.Bd3 Bd6 7.O-O O-O 8.c4 Nc6 9.Nbd2 Ng4 10.Bxh7+ Kh8 11.Ng5 Bxh2+ 12.Kh1 e5 13.Rxf7 exd4 14.Qf3 Re8 15.Rf1 Bd7 16.Rxg7 Kxg7 17.Qf7+ Kh8 18.Qh5 Kg7 19.Qg6+ 1-0, Diemer - von Freytag, 1954 (:-).

4.Nd2 Nf6

A) 5.c4

B

B) 5.f3

C) 5.Ne2 b6 6.Ng3 (6.a3 Bb7 7.c4 c5 8.dxc5 Bxc5 9.Bxc5 bxc5 10.Nc3 O-O -/+ [ECO]) 6...Bb7 7.c4 Be7 (7...Nbd7 8.Qc2 Bb4 -/+ [Pachman]) 8.Be2 O-O 9.f3 exf3 10.Bxf3 Bxf3 11.Qxf3 Na6 12.O-O c5, 0-1 in 28, Zaugg - Buffat, 1991.

4.f3

4.f3?! most likely transposes to other variations. Besides, Black has clear advantage after 4...Bb4+ 5.Nc3 c5!. I have only "studied" the following original line:

4...f5 5.fxe4?!

4...Bd6!? 5.Be3 Nh6 6.Bxh6 Qh4+ 7.Ke2 exf3+ 8.Nxf3 Qxh6, 0-1 in 36, Brause - Fuzzy, FICS 1996. Here Fuzzy, another program, is more innovative; its rating is over 2400.

Various

In general, other (good) moves transpose to more common variations.