Does this do what you want?
bwplot(<...>,
scales = list(x = list(alternating = 1, tck = c(1, 0))))
This is described in detail under 'scales' in ?bwplot.
Not quite. This controls, if I've understood it correctly, the axis tick marks and such (I also tried the argument "draw=FALSE" in the list above).
The analogue I have in mind is the simple plot axes control like this
plot(<...>, axes=FALSE)
axis(1)
which would only draw the bottom axis, with tick marks and label. Those arguments, of course, don't work for the lattice graphics, but I did notice a similar control in the panel.aixs() function, but I just don't know how to get to it.