Broadband isothermal dielectric relaxation measurements of anhydrous fructose, glucose, galactose, sorbose, and ribose were made at ambient pressure in their liquidus and glassy states. We found a new secondary relaxation in fructose and glucose that is slower than those seen before by others. This new secondary relaxation also appears in the dielectric spectra of galactose, sorbose, and ribose, and hence it is a general feature of the relaxation dynamics of the monosaccharides. Dielectric measurements at elevated pressure of fructose and ribose show that the new secondary relaxation shifts to lower frequencies with applied pressures, mimicking the behavior of the alpha-relaxation. In contrast, the faster secondary relaxation remains stationary on applying pressure. These results together with other inferences indicate that the slower secondary relaxation bears relations to the alpha-relaxation, and hence, it is the true Johari-Goldstein secondary relaxation of the monosaccharides.