A class of quadratically convergent regula falsi iterative methods for solving nonlinear equations f(x)=0 is proposed in Chen and Li (Appl. Numer. Math. 57:80–88, 2007). It is also shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of Chen and Li (Appl. Numer. Math. 57:80–88, 2007) by a function p(x) defined suitably. A convergence theorem for establishing the cubic convergence of both the sequence of iterates and the sequence of diameters to the root is given. The numerical examples are worked out to demonstrate that our modified methods are more effective and comparable to those given in Chen and Li (Appl. Numer. Math. 57:80–88, 2007) as well as Newton’s method, Steffensen’s method and regula falsi method.