The five elements that are the most common among cycles involving living organisms are carbon, oxygen, nitrogen, sulfur, and phosphorus. They, along with hydrogen, compose about 90% of all the elements in living organisms. Other elements like trace metal ions make up the rest.
The carbon cycle utilizes microbes to convert dead biomass to carbon dioxide. Some microbes can also utilize carbon dioxide to produce carbohydrate food material like plants do.
The nitrogen cycle also utilizes microbes in a similar way. Some microbes facilitate nitrogen fixation where gaseous nitrogen is converted to ammonia (NH3) which then gets oxidized to various nitrates. Other microbes facilitate dentrification, which is the opposite process where oxidized nitrogen is reduced to nitrogen gas (N2).
Sulfur and phosphorus mostly exist in nature in oxidized form (sulfates and phosphates). Microbes return these elements back to the soil from dead biomass.
Oxygen doesn't really have its own cycle but is a part of the elemental cycles for all of the other elements already mentioned. This is because oxygen is an excellent oxidizing agent and can readily be used in a variety of chemical reactions in nature. The process of photosynthesis in cyanobacteria and algae produces most of the oxygen gas (O2) in the planet's atmosphere as a byproduct.