The Karlin-McGreogor diagonalization can be used to answer recurrence/transience questions, as well as those of probability harmonic functions, occupation times and hitting times, and a large number of other quantities obtained by solving various recurrence relations, in the study of Markov chains. However with some exceptions (see Karlin-McGreogor 1975) those were nearest neighbor Markov chains on half-line. Grünbaum (2007) mentions two main drawbacks to the method as (a) «typically one cannot get either the polynomials or the measure explicitly», and (b) «the method is restricted to “nearest neighbour” transition probability chains that give rise to tridiagonal matrices and thus to orthogonal polynomials». In this talk we will give possible answers to the second question of Grünbaum for general reversible Markov chains. In addition, we will consider possible applications of the newer methods in orthogonal polynomials such as using Riemann-Hilbert approach, and their probabilistic interpretations.