Quintessential dark energy with density ρ and pressure p is governed by an equation of state of the form p=ωqρ with the quintessential parameter ω_qin (-1;-1/3). We derive the geometry of quintessential rotating black holes, generalizing thus the Kerr spacetimes. Then we study the quintessential rotating black hole spacetimes with the special value of ωq = -2/3 when the resulting formulae are simple and easily tractable. We show that such special spacetimes can exist for the dimensionless quintessential parameter c < 1/6 and determine the critical rotational parameter a0 separating the black hole and naked singularity spacetime in dependence on the quintessential parameter c . For the spacetimes with ωq = -2/3 we give all the black hole characteristics and demonstrate local thermodynamical stability. We present the integrated geodesic equations in separated form and study in details the circular geodetical orbits. We give radii and parameters of the photon circular orbits, marginally bound and marginally stable orbits. We stress that the outer boundary on the existence of circular geodesics, given by the so-called static radius where the gravitational attraction of the black hole is balanced by the cosmic repulsion, does not depend on the dimensionless spin of the rotating black hole, similarly to the case of the Kerr-de Sitter spacetimes with vacuum dark energy. We also give restrictions on the dimensionless parameters c and a of the spacetimes allowing for existence of stable circular geodesics. Finally, using numerical methods we generalize the discussion of the circular geodesics to the black holes with arbitrary quintessential parameter ωq.