  Long gamma-ray bursts (GRBs) are likely associated with the collapse of
massive stars, which produce dust and are born in dusty environments.
Absorption and scattering of ultraviolet/X-ray photons from the prompt, optical
flash and afterglow emission of the GRB produce dust echoes. We perform
time-dependent calculations of these echoes, accounting for the evolution of
the dust grain distribution due to selective grain destruction by the GRB
radiation, and for off-axis beaming. We explore cloud configurations of
differing density and size -- the echo light curve and spectrum depend on the
cloud radius, with larger clouds peaking at longer wavelengths. For a region ~
3 pc in size with hydrogen density ~ 1000 per cubic centimeter, the echo
spectrum peaks at ~ 3.6 microns and ~ 8.8 eV for thermal and scattered
components, respectively. Dust echoes should be detectable with the Very Large
Telescope up to z ~ 0.1, IRAC onboard the Spitzer Space Telescope up to z ~
0.2, and NICMOS onboard the Hubble Space Telescope up to z ~ 0.3. Furthermore,
the shape of the echo light curve allows one to infer: the jet opening angle;
the inclination of the jet axis with respect to the line of sight; the size of
the dust-emitting region. For sources with symmetric, bipolar jets, dust echoes
exhibit two bumps in the light curve, making them easily distinguishable from
the rebrightening due to an underlying supernova.
