  Ionizing stellar photons heat the upper regions of planetary atmospheres,
driving atmospheric mass loss. Gas escaping from several hot, hydrogen-rich
planets has been detected using UV and X-ray transmission spectroscopy. Because
these planets are tidally locked, and thus asymmetrically irradiated, escaping
gas is unlikely to be spherically symmetric. In this paper, we focus on the
effects of asymmetric heating on local outflow structure. We use the Athena
code for hydrodynamics to produce 3D simulations of hot Jupiter mass loss that
jointly model wind launching and stellar heating via photoionization. Our
fiducial planet is an inflated, hot Jupiter with radius $R_p=2.14 R_{\rm Jup}$
and mass $M_p = 0.53 M_{\rm Jup}$. We irradiate the initially neutral, atomic
hydrogen atmosphere with 13.6 eV photons and compute the outflow's ionization
structure. There are clear asymmetries in the atmospheric outflow, including a
neutral shadow on the planet's nightside. Given an incident ionizing UV flux
comparable to that of the Sun, we find a steady-state mass loss rate of
~$2\times10^{10}$ g s$^{-1}$. The total mass loss rate and the outflow
substructure along the substellar ray show good agreement with earlier 1D
models, for two different fluxes. Our 3D data cube can be used to generate the
outflow's extinction spectrum during transit. As a proof of concept, we find
absorption of stellar Lyman-alpha at Doppler-shifted velocities of up to $\pm
50$ km s$^{-1}$. Our work provides a starting point for further 3D models that
can be used to predict observable signatures of hot Jupiter mass loss.
