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The transverse motions of nearby dwarf spheroidal (dSph) galaxies contributeline-of-sight components that increase with angular distance from the dSphcenters, inducing detectable gradients in stellar redshift. In the absence ofan intrinsic velocity gradient (e.g., due to rotation or streaming), anobserved gradient in the heliocentric rest frame (HRF) relates simply to adSph's systemic proper motion (PM). Kinematic samples for the Milky Way'sbrightest dSph satellites are now sufficiently large that we can use stellarredshifts to constrain systemic PMs independently of astrometric data. Datafrom our Michigan/MIKE Fiber System (MMFS) Survey reveal significant HRFvelocity gradients in Carina, Fornax and Sculptor, and no significant gradientin Sextans. Assuming there are no intrinsic gradients, the data provide arelatively tight constraint on the PM of Fornax,(mu_{alpha}^{HRF},mu_{delta}^{HRF})=(+48 +/- 15,-25 +/- 14) mas/century, thatagrees with published HST astrometric measurements. Smaller data sets yieldweaker constraints in the remaining galaxies, but our Carina measurement,(mu_{alpha}^{HRF},mu_{delta}^{HRF})=(+25 +/- 36,+16 +/- 43) mas/century, agreeswith the published astrometric value. The disagreement of our Sculptormeasurement, (mu_{alpha}^{HRF},mu_{delta}^{HRF})= (-40 +/- 29, -69 +/- 47)mas/century, with astrometric measurements is expected if Sculptor has arotational component as reported by Battaglia et al. (2008). For Sextans, whichat present lacks an astrometric measurement, we measure(mu_{alpha}^{HRF},mu_{delta}^{HRF})=(-26 +/- 41, +10 +/- 44) mas/century.

Systemic Proper Motions of Milky Way Satellites from Stellar Redshifts: the Carina, Fornax, Sculptor and Sextans Dwarf Spheroidals