at first, i didn´t see the difference between 1. Rd7 and 1. Re7, so i only accidently got it right.
I missed, that the correct reply to 1. .. Qa1 is 2. Nd4 . I thought 2. Qe4, concentrating on moves like 2. .. Qa8+ (3. Qxa8#) or release of black´s bishop on move two, which would allow either 3. Rd8# or 3. Re8#, but of course after 2. Qe4 Qg7 (how did i miss that?) nothing works, but after 2. Nd4 only rook on d7 guarantees mate in three (2. ..Qxd4 3. Qa8#).
Nice.

what problem Rf7?

..then probably 1. .. Qa1
2. Nd4 Qxd4
3. Qa8+ Qd8

1. Ra7 f3
2. Qg6 Qc7

?

I know the problem is to find mate in 3, but in a real game which is "faster": taking 40 minutes to find mate in 3 with 1. Rd7 or taking 3 seconds to play 1. Qh2+ Nh3 2. Qh3+ Qh5 3. Qh5+ Kg8 4.Qh7 mate.

Yes, that seems to explain why 1.Ra7 doesn't work but there's still Re7?

Rd7 and Rf7 - black play Nf3

no checkmate 3 moves

socializa, Nf3 doesn't defend against Rf7. White mates in 3 with 1. Rf7 Nf3 2. Qg2 threatening 3. Rh7#.
If Black defends with 2....Ng5 or 2....Qh5 or 2....Qc7 then 3.Rf8#; and if 2....Qg7 then 3.Qxg7#.

@socializa: 1. Rf7 Qd5
2. Qxd5 Kg8
3. Rf8+ Kh7
(not checkmate)

@paviland: i wrote about 1. Re7 (look at comment no. 8)

1. Rd7 Nf3
2. Qg6
a) 2. .. Ng5 3. Rd8#
b) 2. .. (any queen move) 3. mate with queen on h7 or g7
c) 2. .. (releasing bishop) 3. mate with rook on h7 or d8

socializa, I meant 2. Qg6. Sorry for the mistake.
The mate is similar to what zwitscher-tron described in his most recent post (except d8?), though I built my cases around the threat of 3.Rh7# and not what is moving. As a list it would be:
a) if Black does not defend the h7 square, 3.Rh7#
b) Black defends h7 with 2....Qc7, 2....Qh5, or 2....Ng5, then 3.Rf8#
c) Black defends h7 with 2....Qg7, then 3. Qxg7#

Of course, there are multiple mates available after a number of moves. This division is just what seems most natural to me.

1. Ra7 looked like being a delightful solution. Unfortunately after 1....f3, White must move the queen. He can safely do so because he can no longer be checked by the black queen, but after 2.Qg6, Qc7 White's rook is blocked and there is no mate in 3.

It took a little longer to figure out why 1. Re7 or Rf7 doesn't work; the R must be in position to block the Q after it captures White's N on d4.

A nice try is 1. Qc6, thwarted by 1...Qe4+!. 2. Qxe4 and White's own pieces prevent the Q from reaching the 8th rank to give mate.

In a real game I probably would have played 1. Rb5, either winning the Q or allowing the R to give check on h5, leading to a fairly quick mate though not in three moves.