Single-carrier frequency division multiple access (SC-FDMA) has been selected as the key air interface for uplink transmission in long-term evolution (LTE) specifications. This paper proposes two kinds of minimum mean square error (MMSE)-based equalizers to compensate for the inter-carrier interference (ICI), resulting from the Doppler effect in multipath time-variant channels. The equalizers are designed from the multi-user (MU) and single-user (SU) perspectives. The MU-MMSE equalizer can achieve better performance than the SU-MMSE equalizer, but it has higher computational complexity. Alternatively, the SU-MMSE equalizer which applies the band-limited structure of the ICI matrix is proposed to efficiently reduce the computational complexity at the sacrifice of slight performance degradation on bit error rate (BER). We also develop an iterative receiver by incorporating the ICI cancellation scheme into the SU-MMSE equalization scheme to enhance the BER performance, and propose an interweaving chunk assignment scheme for mobile users with different vehicle speeds. Simulation results show that the proposed scheme can effectively improve the LTE uplink SC-FDMA system performance in multipath time-variant channels.