The combinatorics of the Genocchi numbers was developed by Dumont and various co-authors in the 70's and 80's. More recently, Bousquet-Melou, Claesson, Dukes, Kitaev and Parviainen showed that the 2+2-free posets are in bijection with so-called ascent sequences and with non-negative integer valued upper triangular matrices which have no zero rows or columns. We will show how the Genocchi numbers can be interpreted as the number of up-down ascent sequences thus connecting these various classes of combinatorial objects.