as i noticed in my games it seems clear to me that you should take care of following in amazon chess:

1) weak points h3/h6 in flank openings
In flank openings (kings fianchetto; kings indian and so on) it is obviously that the classical battery of Bishop e3+Amazon d2 deadly points to point h6 (and black vice versa). i won some games with this strategy.

2) weak pawns h3/h6
i think you must take care with moves as h2-h3 or h7-h6. in classical castling positon a sacrifice of a bishop/knight and then the upcoming amazon often leads to the single mate by the amazon (as we discussed)

3) exchanging the amazons
exchanging the amazons leads to a "normal" chess, and then the stronger chess player is in advantage. in my opinion weaker chess players should not exchange the amazons in general. one example is one game form me against chessik - i think my opponent made a mistake to exchange the amazons.

4) get out the amazons in the opening
in chess there is a rule in opening - do not let out your queen fast! i think in amazon you cannot say that belonging to the amazon, because the amazon can threat much more than a normal queen.