  We report an investigation of the London penetration depth $\Delta\lambda(T)$
on single crystals of the layered superconductor Ta$_4$Pd$_3$Te$_{16}$, where
the crystal structure has quasi-one-dimensional characteristics. A linear
temperature dependence of $\Delta\lambda(T)$ is observed for $T\ll T_c$, in
contrast to the exponential decay of fully gapped superconductors. This
indicates the existence of line nodes in the superconducting energy gap. A
detailed analysis shows that the normalized superfluid density $\rho_s(T)$,
which is converted from $\Delta\lambda(T)$, can be well described by a multigap
scenario, with nodes in one of the superconducting gaps, providing clear
evidence for nodal superconductivity in Ta$_4$Pd$_3$Te$_{16}$.
