In the regret-based formulation of Multi-armed Bandit (MAB) problems, except in rare instances, much of the literature focuses on arms with i.i.d. rewards. In this paper, we consider the problem of obtaining regret guarantees for MAB problems in which the rewards of each arm form a Markov chain which may not belong to a single parameter exponential family. To achieve a logarithmic regret in such problems is not difficult: a variation of standard Kullback-Leibler Upper Confidence Bound (KL-UCB) does the job. However, the constants obtained from such an analysis are poor for the following reason: i.i.d. rewards are a special case of Markov rewards and it is difficult to design an algorithm that works well independent of whether the underlying model is truly Markovian or i.i.d. To overcome this issue, we introduce a novel algorithm that identifies whether the rewards from each arm are truly Markovian or i.i.d. using a total variation distance-based test. Our algorithm then switches from using a standard KL-UCB to a specialized version of KL-UCB when it determines that the arm reward is Markovian, thus resulting in low regrets for both i.i.d. and Markovian settings.