  We give an expression for number of points for the family of Dwork K3
surfaces $$X_{\lambda}^4: \hspace{.1in} x_1^4+x_2^4+x_3^4+x_4^4=4\lambda
x_1x_2x_3x_4$$ over finite fields of order $q\equiv 1\pmod 4$ in terms of
Greene's finite field hypergeometric functions. We also develop hypergeometric
point count formulas for all odd primes using McCarthy's $p$-adic
hypergeometric function. Furthermore, we investigate the relationship between
certain period integrals of these surfaces and the trace of Frobenius over
finite fields. We extend this work to higher dimensional Dwork hypersurfaces.
