We present the results of a multi-site photometric campaign to monitor T Tauri stars in the Taurus-Auriga stellar formation region. The campaign - dubbed COYOTES, for Coordinated Observations of Young Stellar ObjecTs from Earthbound Sites - yielded UBVRI magnitudes of 13 classical T Tauri stars, 10 weak-line T Tauri stars, and one suspected Ae Herbig star during 3 months between November 1990 and February 1991. This database is well suited to search for periodic variations from about 1 day up to around 6 weeks, significantly extending the range of timescales properly sampled for T Tauri stars. We find evidence for periodic light variability in all 24 stars of our sample. More precisely, photometric periods are detected in 18 stars at or above the 90% probability level in several photometric bands simultaneously (BD+24°676, DF Tau, DG Tau, DI Tau, DK Tau, DR Tau, GG Tau, GK Tau, IP Tau, RY Tau, GM Aur, LkCa-15, IW Tau, LkCa-4, LkCa-7, LkCa-19, TAP 26, TAP 40), while probable periods, which require confirmation, are reported for 4 other stars (DE Tau, LkCa-21, TAP 41, TAP 57NW), and upper limits on the photometric period are derived for the 2 remaining stars (SU Aur, TAP 9). Light curve analysis and the origin of the variations are discussed in a companion paper (Bouvier et al. 1993, Paper 2).