For a given graph F we consider the family of (finite) graphs G with the Ramsey property for F, that is the set of such graphs G with the property that every two-coloring of the edges of G yields a monochromatic copy of F. For F being a triangle Friedgut, Rodl, Ruciski, and Tetali (2004) established the sharp threshold for the Ramsey property in random graphs. We present a simpler proof of this result which extends to a more general class of graphs F including all cycles. The proof is based on Friedgut's criteria (1999) for sharp thresholds and on the recently developed container method for independent sets in hypergraphs by Saxton and Thomason and by Balogh, Morris and Samotij. The proof builds on some recent work of Friedgut et al. who established a similar result for van der Waerden's theorem.