The invention relates to the alfalfa variety designated 49H344. Provided by the invention are the seeds, plants and derivatives of the alfalfa variety 49H344. Also provided by the invention are tissue cultures of the alfalfa variety 49H344 and the plants regenerated therefrom. Still further provided by the invention are methods for producing alfalfa plants by crossing the alfalfa variety 49H344 with itself or another alfalfa variety and plants produced by such methods.
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