Olfactory navigation is one of the most primitive mechanisms of exploration used by organisms. Navigation by machine olfaction (artificial smell) is a very difficult task to both simulate and solve. With this work, we define olfactory inertial odometry (OIO), a framework for using inertial kinematics, and fast-sampling olfaction sensors to enable navigation by scent analogous to visual inertial odometry (VIO). We establish how principles from SLAM and VIO can be extrapolated to olfaction to enable real-world robotic tasks. We demonstrate OIO with three different odour localization algorithms on a real 5-DoF robot arm over an odour-tracking scenario that resembles real applications in agriculture and food quality control. Our results indicate success in establishing a baseline framework for OIO from which other research in olfactory navigation can build, and we note performance enhancements that can be made to address more complex tasks in the future.