Intermediate-resolution (R~7000) spectroscopy is presented for 76 photometrically selected very low-mass (0.04<M<0.3M_{sun}_) candidate members of the young cluster around sigma Orionis (sigma Ori). More than two-thirds appear to be genuine cluster members on the basis that they exhibit Li I 6708{AA} absorption, weak Na I 8183/8195{AA} features and a radial velocity consistent with the cluster mean. Photometric selection alone therefore appears to be very effective in identifying cluster members in this mass range. Only six objects appear to be certain non-members; however, a substantial subset of 13 candidates have ambiguous or contradictory indications of membership and lack Li absorption. Together with an observed spread in the equivalent width of the Li absorption feature in the cooler stars of our sample, this indicates that there may be deficiencies in our understanding of the formation of this line in cool, low-gravity objects.