  Of the approximately 350 extrasolar planets currently known, of order 10%
orbit evolved stars with radii R >~ 2.5 R_sun. These planets are of particular
interest because they tend to orbit more massive hosts, and have been subjected
to variable stellar insolation over their recent histories as their primaries
evolved off the main sequence. Unfortunately, we have limited information about
the physical properties of these planets, as they were all detected by the
radial velocity method and none have been observed to transit. Here we evaluate
the prospects for detecting transits of planetary companions to giant stars. We
show that several of the known systems have a priori transit probabilities of
>~ 10%, and about one transiting system is expected for the sample of host
stars with R >= 2.5 R_sun. Although the transits are expected to have very
small amplitudes (~few x 10^-4) and long durations (>~ 50 hrs), we argue that
the difficulty with detecting these signals in broadband light is one of
systematic errors and practicality rather than photon noise, even for modest
aperture ~1m telescopes. We propose a novel method that may overcome these
difficulties, which uses narrow-band measurements to isolate the thin ring of
chromospheric emission expected at the limb of giant stars. The transit signals
in these narrow bands are expected to be larger in magnitude and briefer in
duration than in broad-band emission, and thus alleviating many of the
difficulties with transit detection in broad-band emission. Finally, we point
out that it may be possible to discover planetary companions to giant stars
using Kepler, provided that a sufficient number of such targets are monitored.
