An　orientation　of　a　simple　graph　is　referred　to　as　an　oriented　graph.　Caccetta　and　Haggkvist　conjectured　that　any　digraph　on　n　vertices　with　minimum　outdegree　d　contains　a　directed　cycle　of　length　at　most　inverted　right　perpendicularn/dinverted　left　perpendicular.　In　this　paper,　we　consider　short　cycles　in　oriented　graphs　without　directed　triangles.　Suppose　that　alpha(0)　is　the　smallest　real　such　that　every　n-vertex　digraph　with　minimum　outdegree　at　least　alpha(0)n　contains　a　directed　triangle.　Let　epsilon　＜　(3　-　2　alpha(0))/(4　-　2　alpha(0))　be　a　positive　real.　We　show　that　if　D　is　an　oriented　graph　without　directed　triangles　and　has　minimum　outdegree　and　minimum　indegree　at　least　(1/(4　-　2　alpha(0))+epsilon)|D|,　then　each　vertex　of　D　is　contained　in　a　directed　cycle　of　length　l　for　each　4　＜=　l　＜　(4　-　2　alpha(0))epsilon|D|/(3　-　2　alpha(0))　+　2.