We have measured the temperature dependence of the proton spin-lattice relaxation rate R at 8.50 and 22.5 MHz in solid 1,3,5-tri-ethyl-benzene and solid 1,2,4-tri-ethyl-benzene. Analysis of the data strongly suggests that we are studying amorphous states in these slowly solidified organic solids (that are liquids at room temperature). The ethyl groups are static on the Larmor frequency time-scale. There are no simple-model interpretations of the data, but a reasonable model for the dominantly-occurring amorphous state data observed with 1,3,5-tri-ethyl-benzene suggests that two of the three methyl groups are reorienting and the third is static on the proton Larmor frequency time scale. The same approach for the two amorphous states observed in 1,2,4-tri-ethyl-benzene suggests that all three methyl groups are reorienting in one state and that three of the six methyl groups in each pair of molecules are turned off in a second state. We discuss that, whereas specific dynamical statements are model dependent, the proton spin relaxation technique does make some general qualitative statements about the mesostructure of the solid.