  We examine an Unruh-DeWitt particle detector that is coupled linearly to the
scalar density of a massless Dirac field in Minkowski spacetimes of dimension
$d\ge2$ and on the static Minkowski cylinder in spacetime dimension two,
allowing the detector's motion to remain arbitrary and working to leading order
in perturbation theory. In $d$-dimensional Minkowski, with the field in the
usual Fock vacuum, we show that the detector's response is identical to that of
a detector coupled linearly to a massless scalar field in $2d$-dimensional
Minkowski. In the special case of uniform linear acceleration, the detector's
response hence exhibits the Unruh effect with a Planckian factor in both even
and odd dimensions, in contrast to the Rindler power spectrum of the Dirac
field, which has a Planckian factor for odd $d$ but a Fermi-Dirac factor for
even~$d$. On the two-dimensional cylinder, we set the oscillator modes in the
usual Fock vacuum but allow an arbitrary state for the zero mode of the
periodic spinor. We show that the detector's response distinguishes the
periodic and antiperiodic spin structures, and the zero mode of the periodic
spinor contributes to the response by a state-dependent but well defined
amount. Explicit analytic and numerical results on the cylinder are obtained
for inertial and uniformly accelerated trajectories, recovering the $d=2$
Minkowski results in the limit of large circumference. The detector's response
has no infrared ambiguity for $d=2$, neither in Minkowski nor on the cylinder.
