As a Tarrasch adept (who wasn't to shy to state that it was simply better than 4...Nf6), I'm a lifelong addict of 4...Qe7. Actually, I've learned to throw in the flexible 4...Bb6 first (like Alekhine), waiting for the modern piano.

Following Euwe, I prefer the flexible (always make moves you have to do anyway first) 6...d6 over the immediate 6...Nf6 (after 4.c3 Bb6 5.d4 Qe7 6.0-0) one of the points being that the popular 7.h3 can be answered with the paradoxical looking 7...Qf6!? (Van Geet),the idea being that knight manoeuvres like Ng8-e7-g6 (xf4, remember that weakened square with h2-h3) become possible once the position is closed up. Do remember to throw in h7-h6 at some point, not to lose the Q to Bg5. Tactically things work also after say 8.Bg5 Qg6 9. de5 h6 or even 9...Bh3.

Estrins gambit is not really a threat, but tricky to counter.  The thing to remember for black is to keep that pawn-chain flexible: don't let the Q-side get locked up with a2-a4-a5, giving back the pawn with a timely d6-d5 is always an option; My rule of thumb is to take with the Q on d6 (6.d5 Nd8 7.d6), except when an aware white player has thrown in the dangerous Bc1-g5 before making the sac (say 7.Bg5 Nf6 8.d6 cd6 9. Bd5 Ne6! hitting the B, not allowing another doubled pawn on f6).   

Anyway, 4...Nf6 is a lot easier to learn to play well; But, on the plus side,  I've always noted that white players ready to flick out a ton of theory are thrown off course when meeting the Qe7 idea. It does bring about a totally different type of game.

Furthermore, it's a good idea against this old/new (?!) Koltanowski gambit (3.Bc4 Bc5 4.0-0 Nf6 5.d4): Armed with the ...Qe7, one can simply meet 4.0-0 with 4...d6, tranposing and avoiding the Estrin line after 5.c3 Bb6 etc.

Hope this will help

After thinking a bit i am going to contraddict myself 
To my limited understanding the basic idea behind the d3 variation seems to be to play 0-0,Nd2 and then d4, avoiding the annoying reply Bb4+.
The obvious problem is that this is very slow and black can counter in various ways,often leading to static play.
Qe7 might look premature in this setup but if play goes

4...Qe7 5d3 d6 6Nd2 Bb6!? 70-0 Nf6 8d4?! or something similar white seems simple a tempo down on d4 lines.
So Qe7 still stops the basic idea behind 5d3.
Surely i'm a weak player and i'm certainly missing something: how can white change course to seek an advantage?

In the most recent source about the Italian Game, "Beating 1 e4 e5" by John Emms, the author recommends to stay away from the tempting pawn sacrifice with d5-d6. He suggests to continue development with 5.d4 Bb6 6.0-0 Nf6 7.Re1 d6 8.h3 or in "modern" style with d3 instead of the immediate d4, since the queen may feel slightly misplaced on e7 in the resulting positions.

While a solid idea, I think the normal 4... Nf6 is more flexible for Black than commiting the queen so early. 

I recall that in one of the three volumes of Alekhine's best games, he analyzes one that he played this way against Tarrasch.  4...Qe7 is solid and has its theory; I've seen it called the "Strong Point Variation."  I'd be surprised if there aren't quite a few games with it in standard data bases (I'm away from mine right now, visiting relatives up in Maine).