We construct a new family of strong linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al., MIMS EPrint 2016.51, and the new linearizations of polynomial matrices introduced by Faßbender and Saltenberger, Linear Algebra Appl., 525 (2017). In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al., MIMS EPrint 2016.51, allows us to use essentially all the strong linearizations of polynomial matrices developed in the last fifteen years to construct strong linearizations of any rational matrix by expressing such matrix in terms of its polynomial and strictly proper parts.