Tamper-respondent assemblies and methods of fabrication are provided which include a multilayer circuit board, a tamper-detection sensor, and a vent structure. The tamper-detection sensor is embedded within the multilayer circuit board, and defines, at least in part, a secure volume associated with the multilayer circuit board. The vent structure is incorporated into the multilayer circuit board, and includes at least one vent channel. The vent channel(s) is in fluid communication with a space within the secure volume to facilitate venting the space of the secure volume. The space within the secure volume may accommodate, for instance, one or more electronic components to be protected, and the at least one vent channel may, for instance, allow air pressure within the space of the secure volume to equalize with air pressure external to the tamper-respondent assembly.