We investigate the nonequilibrium spin polarization due to a temperature gradient in antiferromagnetic insulators, which is the magnonic analogue of the inverse spin-galvanic effect of electrons. We derive a linear response theory of a temperature-gradient-induced spin polarization for collinear and noncollinear antiferromagnets, which comprises both extrinsic and intrinsic contributions... We apply our theory to several noncentrosymmetric antiferromagnetic insulators, i.e., to a one-dimensional antiferromagnetic spin chain, a single layer of kagome noncollinear antiferromagnet, e.g., $\text{KFe}_3(\text{OH})_6(\text{SO}_4)_2$, and a noncollinear breathing pyrochlore antiferromagnet, e.g., LiGaCr$_4$O$_8$. The shapes of our numerically evaluated response tensors agree with those implied by the magnetic symmetry. Assuming a realistic temperature gradient of $10 \text{K}/\text{mm}$, we find two-dimensional spin densities of up to $\sim 10^6\hbar/\text{cm}^2$ and three-dimensional bulk spin densities of up to $\sim 10^{14}\hbar/\text{cm}^3$, encouraging an experimental detection. (read more)