We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO (SWENO) scheme. The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism. The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENO scheme efficiently. Besides, a scaling technique is used to circumvent the growth of condition numbers as mesh refined. However, weak oscillations still appear when the SWENO scheme deals with complex low density equations. In order to guarantee the maximum-principle-preserving (MPP) property, we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy. A novel procedure is designed to prove this property theoretically. Finally, numerical examples for one- and two-dimensional problems are presented to verify the good performance, maximum principle preserving, essentially non-oscillation and high resolution of the proposed scheme.