We report an investigation of the London penetration depth [Formula: see text] on single crystals of the layered superconductor Ta4Pd3Te16, where the crystal structure has quasi-one-dimensional characteristics. A linear temperature dependence of [Formula: see text] is observed for [Formula: see text], in contrast to the exponential behavior 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 [Formula: see text], which is converted from [Formula: see text], can be well described by a multigap scenario, with nodes in one of the superconducting gaps, providing clear evidence for nodal superconductivity in Ta4Pd3Te16.