An electronic device may be provided with a printed circuit board having padded through-holes. The padded through-holes may be formed from openings in a printed circuit board substrate and elastomeric members in the openings. The elastomeric members may be conductive elastomeric members such as electrically or thermally conductive elastomeric members. The printed circuit board may be secured within a housing for the electronic device using engagement members that extend through padded through-holes. The engagement members may engage with the housing or with additional engagement members that are attached to the housing. The electronic device may include a cowling structure formed over electronic components on a surface of the printed circuit board. The cowling structure may be secured to the printed circuit board using attachment members that engage with the engagement members in the padded through-holes.