This paper presents a fixturing strategy for regrasping that does not require a physical fixture. To regrasp an object in a gripper, a robot pushes the object against external contact/s in the environment such that the external contact keeps the object stationary while the fingers slide over the object. We call this manipulation technique fixtureless fixturing. Exploiting the mechanics of pushing, we characterize a convex polyhedral set of pushes that results in fixtureless fixturing. These pushes are robust against uncertainty in the object inertia, grasping force, and the friction at the contacts. We propose a sampling-based planner that uses the sets of robust pushes to rapidly build a tree of reachable grasps. A path in this tree is a pushing strategy, possibly involving pushes from different sides, to regrasp the object. We demonstrate the experimental validity and robustness of the proposed manipulation technique with different regrasp examples on a manipulation platform. Such a fast and flexible regrasp planner facilitates versatile and flexible automation solutions.