We define piecewise semiprime (PWSP) rings R in terms of a set of triangulating idempotents in R. The class of PWSP rings properly contains both the class of semiprime rings and the class of piecewise prime rings. The PWSP property is Morita invariant and it is shared by some important ring extensions. A ring is PWSP if and only if it has a generalized upper triangular matrix representation with semiprime rings on the main diagonal. Another characterization of PWSP rings involves a generalization of the concept of m-systems and is similar to the description of a semiprime ring in terms of the prime radical. Finally we use the PWSP property to determine (right) weak quasi-Baer rings. These are rings in which the right annihilator of every nilpotent ideal is generated as a right ideal by an idempotent.