Here is my first try at a solution. Your idea was a good one, but
bishops are better than rooks, I surmise.
The two pictures here are placed in some distinct parts of the infinite board.
The first just ensures it is White to move (in check), and that White's king
will never play a role, as capturing a black unit, which are nearly stalemated as is,
will release heavy pieces.
alt text http://www.freeimagehosting.net/uploads/3c8e277e7d.jpg alt text http://www.freeimagehosting.net/uploads/72ef1c9b7e.jpg
So White is left to checkmate with the four bishops and pawns.
White threatens checkmate via a check from below on the northwest diagonal,
and Black can only avoid this by moving the bishop northeast some amount.
Upon Black moving this bishop, White then makes the bishop check anyways,
the Black king moves where the Black bishop was, the pawn moves with check,
the Black king again retreats northeast along the diagonal, and then White
alternately moves the dark-square bishops, giving checks until the Black
bishop is reached when it is mate.
The point of this second picture is that White cannot checkmate Black
unless the Black bishop plays a role. Four bishops are not enough to
checkmate a king on an infinite board, and hopefully I have set it up so
that the White pawns play no part once Black starts the king running northeast.
Pawns are not too valuable when they cannot become queens.
In extended chess notation, White plays 1. Ke5 on board A,
then Black plays 1...Bz26 on board B, followed by
2. Bg3+ Kf6 3. e5+ Kg7 3. Bi5+ Kh8 4. Bf10+ Ki9 5. Bk7+ Kj10 6. Bh12+ ...,
as White successively cuts off NW-SE diagonals until the Black bishop
is reached. By moving the bishop X squares northeast on move 1, Black
can delay the checkmate for X moves, if I set this up proper.
Other plans by White should be beatable by moving the Black king off
the long diagonal or capturing the light White bishop with the pawn.
Once Black's king exits the area with the pawns, the Black bishop
must be a part of the mating pattern. I don't think the Black king
can be forced back to that area.
Well, this is a first try.
Best Answer
As far as I know, it is still open whether or not there is any finite number of rooks which can force checkmate. However, this is possible. I answered the question over at math.stackexchange describing a strategy which forces checkmate with 96 rooks. This is certainly not optimal. The strategy I describe is very wasteful, keeping some rooks in place even when they are not actually needed, so it can be improved at the expense of becoming more difficult to describe.