There is a positive solution for the decidability of the mate-in-$n$ version of the problem. 

Many of us are familiar with the *White to mate in 3*
variety of chess problems, and we may consider the natural analogue
in infinite chess. Thus, we refine the *winning-position* problem,
which asks whether a designated player has a winning strategy from
a given position, to the *mate-in-$n$* problem, which asks whether
a designated player can force a win in at most $n$ moves from a
given finite position. (And note that as discussed in Johan
Wästlunds's question [checkmate in $\omega$
moves?](http://mathoverflow.net/questions/63423/checkmate-in-omega-moves), there are finite winning positions in infinite
chess which are not mate-in-$n$ for any finite $n$.) Even so, the
mate-in-$n$ problem appears still to be very complicated, naturally
formulated by assertions with $2n$ many alternating quantifiers:
*there is a move for white, such that for every black reply, there
is a countermove by white,* and so on. Assertions with such quantifier complexity are 
not generally decidable, and one cannot expect to search an
infinitely branching game tree, even to finite depth. So one might
naturally expect the mate-in-$n$ problem to be undecidable.

Despite this, the mate-in-n problem of infinite chess
is computably decidable, and uniformly so. Dan Brumleve, myself and Philipp Schlicht have just submitted
an article establishing this to the [CiE 2012](http://www.mathcomp.leeds.ac.uk/turing2012/WScie12/), and I hope to speak on it there
in June.

  > D. Brumleve, J. D. Hamkins and P. Schlicht, ["The mate-in-*n* problem of infinite
chess is decidable,"](http://boolesrings.org/hamkins/the-mate-in-n-problem-of-infinite-chess-is-undecidable/) 10 pages, [arxiv pre-print](http://arxiv.org/abs/1201.5597), submitted to [CiE 2012](http://www.mathcomp.leeds.ac.uk/turing2012/WScie12/).

  > <b>Abstract.</b> Infinite chess is chess played on an infinite
edgeless chessboard. The familiar chess pieces move about according
to their usual chess rules, and each player strives to place the
opposing king into checkmate. The mate-in-$n$ problem of infinite
chess is the problem of determining whether a designated player can
force a win from a given finite position in at most n moves. A
naive formulation of this problem leads to assertions of high
arithmetic complexity with $2n$ alternating quantifiers---*there is
a move for white, such that for every black reply, there is a counter-move
for white,* and so on. In such a formulation, the problem does not
appear to be decidable; and one cannot expect to search an
infinitely branching game tree even to finite depth. Nevertheless,
the main theorem of this article, confirming a conjecture of the
first author and C. D. A. Evans, establishes that the mate-in-$n$
problem of infinite chess is computably decidable, uniformly in the
position and in $n$. Furthermore, there is a computable strategy
for optimal play from such mate-in-$n$ positions. The proof
proceeds by showing that the mate-in-$n$ problem is expressible in
what we call the first-order structure of chess $\frak{Ch}$, which
we prove (in the relevant fragment) is an automatic structure,
whose theory is therefore decidable. Unfortunately, this resolution
of the mate-in-$n$ problem does not appear to settle the
decidability of the more general winning-position problem, the
problem of determining whether a designated player has a winning
strategy from a given position, since a position may admit a
winning strategy without any bound on the number of moves required.
This issue is connected with transfinite game values in infinite
chess, and the exact value of the omega one of chess $\omega_1^{\rm
chess}$ is not known.

The solution can also be cast in terms of Presburger arithmetic, in a manner close to Dan Brumleve's answer to this question. Namely, once we restrict to a given collecton of pieces $A$, then we may represent all positions using only pieces in $A$ as a fixed-length tuple of natural numbers, and the elementary movement, attack and in-check relations are expressible for this representation in the language of Presburger arithmetic, essentially because the distance pieces---rooks, bishops and queens---all move on straight lines whose equations are expressible in Presburger arithmetic. (There is no need to handle sequence coding in general, since the number of pieces does not increase during play.) Since the mate-in-$n$ problem is therefore expressible in Presburger arithmetic, it follows that it is decidable.