To me, not believing in God is quite logical. I'll try to explain.
First, I'd like to define "God" as "supernatural entity/ies interested in our personal lives". My definition suitably includes the God of Judaism and Christianity, Allah and a number of less popular deities. Thus, if my argument is successful, it'll cover more ground for the same price.
Next, remember that there is no logically conclusive argument for the existence of God. If there were, the discussion would be over. Similarly, it is not possible to prove that God does not exist, or that the hypothetical He does not care about us. It's possible that something very subtle is going on that we are just not sophisticated enough to understand.
So what's left? Failing absolute proof, a common and time-tested technique is Occam's Razor: Examining a set of conflicting claims, the one that requires the least number of improbable assumptions is most likely to be correct. Below, I briefly lay out some assumptions that must be true to support the idea of God, and those that must be true if there is none. Be aware of the instruction manual for the razor, which points out its limitation: It is a tool for choosing the hypothesis most closely aligned with your existing knowledge; it doesn't create Truth out of thin air, but helps you form a plausible opinion.
Suppose there's a God, then. Most religions claim that at one time, thousands to hundreds of years ago, He made himself known to mankind, and performed some miracles while doing so: He flooded the Earth (Noah), stopped the sun (Joshua), cured the blind (Jesus), etc. Now let's consider that some of these miracles were impressive enough that they would be recognized as such worldwide, and that at all the proposed times there were civilizations on hand with sufficient scholarly skills and resources to take note and record these miracles. Yet no ancient Chinese document ever mentions the sun standing still, even though the Chinese bureaucracy was known to record in great detail even the most mundane things. Similarly, no Roman scholar ever saw fit to document the practice of miracles in Jerusalem. What all religious documents have in common is that for all of their more extraordinary claims, they are the only reference! Thus, if God did indeed perform miracles, while on the one hand insisting on being worshipped, He also saw fit to purge the evidence from worldly memory and documentation.
What's the counter-assumption? That the core tenets of contemporary religions are myths, just like other texts that are nowadays taken to be myths. The Egyptian sun-god, the Greek pantheon, the bloodthirsty gods of the Incas were –so we hold today– simply the products of fertile human imagination, intended to hold common people in awe and help control society. Pre-historic spirit worship (still extant in a few "primitive" cultures today), the worship of sun and rain deities, polytheism and monotheism all reflect the world view of the people who created them. Hunter-gatherers worshipped animal spirits; agrarians paid homage to entities who controlled the weather; the god of the Jews, a relatively small group fighting for survival in the Middle East, was a fierce and vindictive warlord.
On to analysis! Contemporary religions raise many questions. In view of conflicting assertions, one of the foremost is which one to believe. Logic dictates that of conflicting claims, only one can be true. Or, taken another way, if there are 10 religions, only one or two can be "correct", and the rest must be myths. Another problem with the literal interpretation of religious texts is that they are often internally inconsistent to boot. There's only so much inaccuracy that can be hand-waved away as "metaphor" or "another way of looking at the same thing." Certainly, contemporary religions admit that their central texts were penned by humans, and inconsistencies can be attributed to their fallibility. But having accepted that the Bible and other texts are human-written, self-contradictory and inaccurate, it's not a big step to assume that humans not only wrote these texts, but are also the sole creators of their fictitious or at least fanciful content. Other easy assumptions are that people want to believe the teachings of religion because of the comfort to be drawn therefrom, while rulers support the perpetutation of religion because it helps control the mob.
Either God is a being who transcends all known laws of nature, once worked miracles but then strove to hide evidence of His existence, and inspired contradictory accounts of His deeds and His will.
Or God is a fiction created and documented by ordinary humans, a heroic father figure reflecting their needs and desires.
I don't know about you. To me, the second assumption is a lot easier because it doesn't call for any extraordinary supporting assumptions. And Occam is on my side.
wertperch is now the second person to point out that God doesn't necessarily have to be a miracle monger. He's right, I admit it. Given the set of all variations on God being worshipped today, that subset which doesn't (according to its followers) claim impressive miracles is harder to dispute by my argument. For them, I'm reduced to talking about how the evolution of religion is a more natural and likely process than the creation (or existence since the beginning of time) of an infinitely supernatural being.
But hey, what kind of wimpy cop-out deity has all the power in the universe and then never shows it off? Seriously, I believe the putative display of power attributed to the god of Judaeo-Christianity is one of the reasons for the smashing (sometimes literally) success of Judaism and later Christianity. Christians are happy to feel loved by their god, but proud to follow a strong, active one who shows His hand - at least historically. Also, the faithful sometimes call on these miracles in a kind of circular proof for the existence of their deity.
I know too little about Islam to even make an educated guess, but as for the billion practicing or at least professing Christians alive today I venture to posit that the majority believe in a God of miracles, and this is the one at whom my argument is directed. Toppling the rest is left as an exercise for the reader.