In this talk we will look at five pencils of projective quartic Delsarte K3 surfaces. Over finite fields, we give explicit formulas for the point counts for each family, written in terms of hypergeometric sums. Over the complex numbers, we match the periods of the corresponding family with hypergeometric differential operators and series. We also obtain a decomposition of the incomplete L-function of each pencil in terms of hypergeometric L-series and Dedekind zeta functions. This gives an explicit description of the hypergeometric motives geometrically realized by each pencil. This is joint work with Rachel Davis, Jessamyn Dukes, Thais Gomes Ribeiro, Eli Orvis, Leah Sturman, and Ursula Whitcher.