Several specific OSHA standards require that employees receive training. In the standards books (29CFR 1910 for General Industry, and 1926 for Construction, for example), the tables of contents do not list general "training" headings. However, in the index, under "Training Personnel," several standards are referenced. Better yet, I recommend that you obtain OSHA's "Training Requirements in OSHA Standards and Training Guidelines." It is an across-the-board, dedicated publication that ostensibly contains every reference to training in the standards.
OSHA-mandated or not, all employees should receive at least basic safety and health training. OSHA-dictated training requirements are not the "end all." You must decide what topics (and to what degree) should be included in orientation programs, even when not legally obligated.