The 3GPP LTE series of standards have made provisions for device-to-device (D2D) and vehicle- to-everything (V2X) communication in recent releases. The interface accommodating such communication is termed the sidelink (SL), which reuses some physical layer specifications originally defined for the uplink, such as single-carrier frequency-division multiple access (SC-FDMA), with some modifications. One modification is the use of doubly dense demodulation reference signals (DMRS) in the V2X setting to assist channel estimation. An SC-FDMA channel estimator typically starts with a least-squares (LS) estimation of the channel frequency response (CFR) at the DMRS. It then obtains CFR estimates for the data subcarriers by some time-frequency filtering. This work considers SL channel estimation techniques. The proposed method employs a novel linear minimum mean-square error (LMMSE) filtering mechanism in the frequency domain to reduce the noise in the LS CFR estimates and then, depending on the DMRS density, uses particularly designed LMMSE filtering or polynomial interpolation in the time domain to obtain the desired CFR estimates for the data subcarriers. Simulation results show that the method can yield superior performance over a wide range of channel conditions.