We consider three-dimensional domino tilings of cylinders D x [0, N], where D [contains] R2 is a fixed quadriculated disk and N [belongs to] N. A domino is a 2x1x1 brick. A flip is a local move in the space of tilings T (D x [0, N]): two adjacent and parallel dominoes are removed and then placed in a different position. The twist is a flip invariant which associates an integer number to each tiling. For certain disks D, called regular, any two tilings of D x [0, N] sharing the same twist can be connected through a sequence of flips when extra vertical space is added to the cylinder. We prove that the absence of a bottleneck in a hamiltonian disk implies regularity. Conversely, we show that the presence of a bottleneck in a disk D often indicates irregularity. In many cases, we further demonstrate that D belongs to a specific class of irregular disks, which we define as strongly irregular. Furthermore, for any strongly irregular disk D, we prove that the connected components under flips consist of exponentially small fractions of T (D x [0, N]).