We report magnetic susceptibility (χ) and heat capacity (Cp) measurements along with ab initio electronic structure calculations on PbCuTe2O6, a compound made up of a three-dimensional (3D) network of corner-shared triangular units. The presence of antiferromagnetic interactions is inferred from a Curie-Weiss temperature (θCW) of about -22 K from the χ(T) data. The magnetic heat capacity Cm data show a broad maximum at Tmax≃1.15 K (i.e., Tmax/θCW≃0.05), which is analogous to the the observed broad maximum in the Cm/T data of a hyper-kagome system, Na4Ir3O8. In addition, Cm data exhibit a weak kink at T*≃0.87 K. While the Tmax is nearly unchanged, the T* is systematically suppressed in an increasing magnetic field (H) up to 80 kOe. For H≥80 kOe, the Cm data at low temperatures exhibit a characteristic power-law (Tα) behavior with an exponent α slightly less than 2. Hopping integrals obtained from the electronic structure calculations show the presence of strongly frustrated 3D spin interactions along with non-negligible unfrustrated couplings. Our results suggest that PbCuTe2O6 is a candidate material for realizing a 3D quantum spin liquid state at high magnetic fields. © 2014 American Physical Society.