We report the first four magnetic representatives of the trigonal layered A(2)M(4+)TeO6 (here, M = Mn) family. Na2MnTeO6 was synthesized from NaMnO2, NaNO3, and TeO2 at 650-720 degrees C, but analogues for which A = Li and K could not be obtained by direct synthesis. However, those for which A = Li, Ag, and TI (but not K) were prepared by exchange reactions between Na2MnTeO6 and the corresponding molten nitrates. The oxygen content was verified by redox titration. According to the X-ray diffraction Rietveld analysis, The four new compounds are isostructural with Na2GeTeO6, trigonal (P (3) over bar 1c), based on ilmenite-like layers of edge-shared oxygen octahedra occupied by Mn(4+) and Te(6+) in an ordered manner. These layers are separated by cations A, also in a distorted octahedral coordination. However, off-center displacement of Tl+ is so strong, due to the lone-pair effect, that its coordination is better described as trigonal pyramid. Each MnO6 octahedron shares two opposite faces with AO(6) octahedra, whereas TeO6 octahedra avoid sharing faces. Besides this double-layered structure, Na2MnTeO6 was often accompanied by a transient triple-layered rhombohedral polytype. However, it could not be prepared as a single phase and disappeared on annealing at 700-720 degrees C. All A(2) MnTeO6 samples (A = Ag, Li, Na, or Tl) revealed the unusual phenomenon of hidden magnetic order. Low-field magnetic susceptibility data exhibit a Curie- Weiss type behavior for all samples under study and do not show any sign of the establishment of long-range magnetic order down to 2 K. In contrast, both the magnetic susceptibility in sufficiently high external magnetic fields and the zero-field specific heat unambiguously revealed an onset of antiferromagnetic order at low temperatures. The frustration index f = Theta/T-N takes values larger than the classical values for three-dimensional antiferromagnets and implies moderate frustration on the triangular lattice.