This answer contains spoilers from ME 1 and 2.
The Reapers do not seek to destroy all organic life, nor do they try. Their interest in organic life seems to lie solely in that they use it for reproduction - i.e., to make more Reapers.
To this end, they seed the galaxy with their artifacts - The Mass Relays and The Citadel itself - such that life will develop in such a way to best serve their inscrutable purposes.
From Mass Effect 1:
Sovereign: Organic life is nothing but a genetic mutation, an accident. Your lives are measured in years and decades. You wither and die. We are eternal, the pinnacle of evolution and existence. Before us, you are nothing. Your extinction is inevitable. We are the end of everything.
Commander Shepard: Whatever your plan is, it's going to fail. I'll make sure of that.
Soverign: Confidence born of ignorance. The cycle cannot be broken.
Squad Member: Cycle? What cycle?
Sovereign: The pattern has repeated itself more times than you can fathom. Organic civilizations rise, evolve, advance, and at the apex of their glory they are extinguished. The Protheans were not the first. They did not create the Citadel. They did not forge the mass relays. They merely found them - the legacy of my kind.
Commander Shepard: Why would you construct the mass relays and leave them for someone else to find?
Sovereign: Your civilization is based on the technology of the mass relays. Our technology. By using it, your civilization develops along the paths we desire. We impose order on the chaos of organic life. You exist because we allow it, and you will end because we demand it.
The Cycle, as mentioned by Sovereign, occurs every 50,000 years (presumably as the Reaper fleet travels between galaxies), and involves the methodical destruction of every space-faring species. If you haven't discovered FTL travel, the Mass Relays, and the Mass Effect, the Reapers pretty much ignore you.
This is why Humans were not wiped out when the Protheans were, despite there being prominent Cro-Magnons rutting about on Earth (the Protheans even had an observation base on Mars specifically to watch Humans). We were effectively "beneath their notice".
This focus on space-faring races is further confirmed in ME3, where Admiral Hackett remarks that the Yahg homeworld is being completely ignored, even intentionally passed over, by the Reaper fleets invading nearby (they haven't achieved space travel). In his words "If we fail, it might be them in our shoes, next time".