Rubuta ci gaba don wannan sakin layi - Iturrizaga ya lashe gasar zakarun kasa hudu a jere daga 2005 zuwa 2008, kuma yana taka leda a Venezuela a gasar Olympics ta Chess tun 2004. Kyakkyawan aikinsa ya zo ne a gasar Olympiad ta Chess ta 37
(2006), lokacin da ya ci 8.5/11 maki wasa na biyu jirgin. Sakamakon ya ba shi lambar tagulla ta mutum. Ya cancanci zuwa gasar cin kofin duniya ta Chess ta 2007 bayan ya raba matsayi na farko a Gasar Cin Kofin Nahiyar Amurka. An saita shi don fuskantar Peter Svidler a zagaye na farko, amma ya rasa wasan farko saboda ɓacewa tsakanin filayen jirgin saman Rasha. Ya rasa wasa na biyu. A shekara ta 2008, Iturrizaga ya lashe wasan neman shiga rukuni na C a gasar cin kofin Chess ta 2009 ta Corus, inda ya doke GM Alexandr Fier da ci 3-1 a wasan karshe. Ya ƙare a matsayi na takwas daga cikin goma sha huɗu a babban gasar, yana cin 5.5/13. Iturrizaga ya gama na biyu a Gasar Ibero-American ta 2008 a Linares, Spain, inda ya yi rashin nasara ga Julio Granda a wasan karshe. Iturrizaga shine mutum na karshe da ya isa gasar, dole ne ya buga wasa daya da daddare, kuma kusan ya makara a daya daga cikin wasannin.