Radio astronomical observations at millimeter and submillimeter wavelengths are a very important tool for astrophysical research. However, there is a huge area in northeastern Eurasia, including the whole Russian territory, which lacks sufficiently large radio telescopes effectively operating at these wavelengths. In this review, we describe our long-term efforts to find suitable sites for such radio telescopes in this area, that is, sites with good atmospheric transparency at millimeter and submillimeter waves. We describe methods and instruments used for measurements and evaluations of the atmospheric opacity. They include special radiometric systems, which are used for estimations of the atmospheric opacity in the transparency windows from the sky brightness measurements. Evaluation of the precipitable water vapor from such measurements by the artificial neural network is discussed. Other approaches use water vapor radiometers, global atmospheric models and signals of the Global Navigation Satellite Systems. To date, long-term radiometric monitoring has been performed at several candidate sites, and atmospheric conditions for many sites have been evaluated using global atmospheric models. Several sites with the best atmospheric transparency at millimeter and submillimeter wavelengths have been selected. They can be effectively used for astronomical observations, at least in the major atmospheric transparency windows at 1.3 mm and 0.85 mm. In general, the results show that northeastern Eurasia is a promising area for submillimeter astronomy. These results can also be used for space communications and radar systems.