Accurate quantification of geocenter motion is important for maintaining the reference frame and estimating large-scale mass variations using the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity solutions. Geocenter motion (equivalent to the variations in the degree-1 spherical harmonics of the gravity field) can be determined from different geodetic techniques and approaches, and the results generally show reasonable agreement, but significant differences still exist. To be more consistent with GRACE gravity solutions, here we present an improved approach to solve geocenter motion using GRACE GPS data. Over the past 15 years, GRACE satellites have acquired enough GPS data for studying the geocenter motion. In the meantime, data processing methods, reference system, and background geophysical models for GRACE precise orbit determination have also been significantly improved. Those aspects are very important for accurate determination of geocenter motion from GRACE GPS observations. For comparison, geocenter motion is also derived from LAGEOS satellite laser ranging (SLR) observations. With these independent geocenter motion solutions from GRACE GPS and LAGEOS SLR, we explore the reasons that lead to the differences between the solutions, and try to resolve these discrepancies. Daily geocenter motion time series from GRACE GPS data and 28-day geocenter variations from LAGEOS SLR observations for the time span 2003-2016 have been derived. Internal comparisons between the GRACE-A and GRACE-B geocenter motion time series and external comparisons between GRACE and LAGEOS show good agreements after using the improved approach. To verify the results, the annual geocenter motion from this study is compared with other recent geocenter motion solutions as well as predictions from geophysical models. The comparisons show reasonable agreements in both amplitude and phase with our improved approach.