This paper is entirely devoted to analyse and solve the H-infinity optimal state feedback sampled-data control design problem for continuous-time Markov jump linear systems. This is an important unsolved problem in the context of optimal control theory. To this end, necessary and sufficient optimality conditions are characterised in terms of a specific nonlinear two-point boundary value problem. A global convergent algorithm able to solve iteratively the optimality conditions is provided. Moreover, some mathematical properties of the solution of a differential Riccati equation are raised in order to translate the previous conditions into linear matrix inequalities. This result allows a mode independent feedback control structure for the Markovian system under analysis. A practical application borrowed from the literature is included for illustration.