Atheists believe the universe is random that it operates and behaves randomly. Only if God exists could the universe not be random, but if the universe really were random we would not be here.
It's difficult to know whether to categorize this claim as a myth or not because atheists do commonly describe the universe as "random" in some sense, but that doesn't make the claim true. Where the problem lies is in how religious theists who make or believe this claim misunderstand or misrepresent what "random" means. When used correctly, this can be a correct description of many atheists' positions. When used in the manner that many theists seem to intend it, it's incorrect and thus a myth.
First, let's take a look at a couple of the basic definitions of random:
1. Proceeding, made, or occurring without definite aim, purpose, reason, or pattern: the random selection of numbers.
2. Statistics: of or characterizing a process of selection in which each item of a set has an equal probability of being chosen; where all outcomes are equally likely, as in the testing of a blood sample for the presence of a substance. Unpredictable.
Atheists may believe that the universe is random in most of the senses described above, depending on the exact context or situation. Atheists obviously don't believe that there are any gods who created and are guiding the universe, so it's unlikely that they believe the universe was made or exists with purpose, aim, or reason. Atheists don't dispute that there are patterns in the universe, though, and even in truly random sets like sequences of numbers patterns will emerge. The existence of patterns does not contradict the existence of randomness.
Atheists may also believe that, in many circumstances, all members of a set of possible outcomes are either equally likely to occur, almost equally likely to occur, or if not then it's not possible to predict which is the most likely to occur. Religious theists may often believe that whatever happens does so as part of God's plans for them and the world, but atheists don't believe in any gods which might be guiding events and causing certain outcomes to occur rather than others. Whatever happens, happens according to natural laws which may or may not favor particular outcomes and sometimes, those outcomes may be random for all practical intents and purposes.
When religious theists accuse atheists of believing in a "random" universe, though, they don't normally seem to mean that atheists simply disbelieve that the universe was created without a specific purpose or that some events are unpredictable. The first is almost the definition of atheism and some theists agree with the latter, so neither of these would be much of a criticism.
Instead, what many religious theists seem to mean is that atheists' belief require a universe without order that randomness entails disorder and that therefore the existence of order in the universe means that some god (preferably their god) must exist. The equation of randomness with disorder is incorrect and this is why the above claim amounts to a myth: atheists' beliefs do not entail that the universe be disordered and order can in fact arise spontaneously, without any directed intervention from a conscious being.
Atheists do not deny that the universe has order, structure, and patterns. What atheists deny, and science supports them in this, is the idea that order, structure, and patterns are incompatible with randomness or that they require the prior presence of some "designer" for them to exist. A structured, ordered universe can be random in some aspects, does not need any sort of god to explain how it originated, and does not need any sort of god to explain why it continues to exist now.