We present the results of a new sensitive survey of neutral hydrogen above and below the Galactic Center with the Green Bank Telescope. The observations extend up to Galactic latitude |b|<10{deg} with an effective angular resolution of 9.5' and an average rms brightness temperature noise of 40mK in a 1km/s channel. The survey reveals the existence of a population of anomalous high-velocity clouds extending up to heights of about 1.5kpc from the Galactic plane and showing no signature of Galactic rotation. These clouds have local standard of rest velocities |V_LSR_|<~360km/s, and assuming a Galactic Center origin, they have sizes of a few tens of parsec and neutral hydrogen masses spanning 10-10^5^M_{sun}_. Accounting for selection effects, the cloud population is symmetric in longitude, latitude, and VLSR. We model the cloud kinematics in terms of an outflow expanding from the Galactic Center and find the population consistent with being material moving with radial velocity V_w_~330km/s distributed throughout a bicone with opening angle {alpha}>140{deg}. This simple model implies an outflow luminosity L_w_>3x10^40^erg/s over the past 10Myr, consistent with star formation feedback in the inner region of the Milky Way, with a cold gas mass-loss rate <~0.1M_{sun}_/yr. These clouds may represent the cold gas component accelerated in the nuclear wind driven by our Galaxy, although some of the derived properties challenge current theoretical models of the entrainment process.