In this paper, formulation of a novel consecutive-interpolation 4-node tetrahedral finite element (CTH4) and its applications to the analysis of heat transfer problems in three-dimension (3D) are presented. The field variables approximation is performed on the way of taking both the nodal values and their averaged nodal gradients into account, in terms of the consecutive-interpolation procedure (CIP). The new CTH4 element proposed inherently possesses many desirable advantages over the conventional tetrahedral element (TH4) such as the higher accuracy, higher-order continuity, and continuous nodal gradients without smoothing operation. Importantly, the number of degrees of freedom of the system does not change, but still remains the nodal values as that of the TH4 element. We demonstrate the accuracy and performance of the developed CTH4 element through a series of numerical experiments of 3D heat transfer problems, in which comparison between the present obtained results, and reference solutions derived from analytical solutions and other numerical approaches is made. We additionally propose a general formulation of auxiliary functions in terms of the CIP method. As a result, a family of CIP-based elements in all dimensions (i.e., 1D up to 3D) can now straightforwardly be estabilshed since any auxiliary functions required by the CIP scheme are easily to be generated by using the present general formulation. (C) 2016 Elsevier Ltd. All rights reserved.