Small planets are common around late-M dwarfs and can be detected through highly precise photometry by the transit method. Planets orbiting nearby stars are particularly important as they are often the best-suited for future follow-up studies. We present observations of three nearby M dwarfs referred to as EIC-1, EIC-2, and EIC-3, and use them to search for transits and set limits on the presence of planets. On most nights our observations are sensitive to Earth-sized transiting planets, and photometric precision is similar to or better than TESS for faint late-M dwarfs of the same magnitude (I ≈ 15 mag). We present our photometry and transit search pipeline, which utilizes simple median detrending in combination with transit least-squares-based transit detection. For these targets, and transiting planets between one and two Earth radii, we achieve an average transit detection probability of ∼60% between periods of 0.5 and 2 days, ∼30% between 2 and 5 days, and ∼10% between 5 and 10 days. These sensitivities are conservative compared to visual searches.