We present new Spitzer Space Telescope observations of two fields in the Orion OB1 association. We report here IRAC/MIPS observations for 115 confirmed members and 41 photometric candidates of the ~10 Myr 25 Orionis aggregate in the OB1a subassociation, and 106 confirmed members and 65 photometric candidates of the 5 Myr region located in the OB1b subassociation. The 25 Orionis aggregate shows a disk frequency of 6% while the field in the OB1b subassociation shows a disk frequency of 13%. Combining IRAC, MIPS and 2MASS photometry we place stars bearing disks in several classes: stars with optically thick disks (class II systems), stars with an inner transitional disks (transitional disk candidates) and stars with evolved disks; the last exhibit smaller IRAC/MIPS excesses than class II systems. In all, we identify 1 transitional disk candidate in the 25 Orionis aggregate and 3 in the OB1b field; this represents ~10% of the disk bearing stars, indicating that the transitional disk phase can be relatively fast. We find that the frequency of disks is a function of the stellar mass, suggesting a maximum around stars with spectral type M0. Comparing the infrared excess in the IRAC bands among several stellar groups we find that inner disk emission decays with stellar age, showing a correlation with the respective disk frequencies. The disk emission at the IRAC and MIPS bands in several stellar groups indicates that disk dissipation takes place faster in the inner region of the disks. Comparison with models of irradiated accretion disks, computed with several degrees of settling, suggests that the decrease in the overall accretion rate observed in young stellar groups is not sufficient to explain the weak disk emission observed in the IRAC bands for disk bearing stars with ages 5 Myr or older.