This work was motivated by previous findings that the measured laser-driven heat front propagation velocity in under-dense TiO2/SiO2 foams is slower than the simulated one [Pérez et al., Phys. Plasmas 21, 023102 (2014)]. In attempting to test the hypothesis that these differences result from effects of the foam microstructure, we designed and conducted an experiment on the GEKKO laser using an x-ray streak camera to compare the heat front propagation velocity in "equivalent" gas and foam targets, that is, targets that have the same initial density, atomic weight, and average ionization state. We first discuss the design and the results of this comparison experiment. To supplement the x-ray streak camera data, we designed and conducted an experiment on the Trident laser using a new high-resolution, time-integrated, spatially resolved crystal spectrometer to image the Ti K-shell spectrum along the laser-propagation axis in an under-dense TiO2/SiO2 foam cylinder. We discuss the details of the design of this experiment, and present the measured Ti K-shell spectra compared to the spectra simulated with a detailed superconfiguration non-LTE atomic model for Ti incorporated into a 2D radiation hydrodynamic code. We show that there is indeed a microstructure effect on heat front propagation in under-dense foams, and that the measured heat front velocities in the TiO2/SiO2 foams are consistent with the analytical model of Gus'kov et al. [Phys. Plasmas 18, 103114 (2011)].